OP you say you are worried about him in general. Is this worries relating to development/ND? (One of my DC really struggled starting primary).
I would expect that moving schools won't fix the problem. It will likely just reoccur in the next setting unless there is something specific about this school which is causing a problem. It's extremely unlikely that the other boys he has palled up with are uniquely terrible - there will be children like them in every school in the country. Mixed age is probably a red herring - this means the school is fairly small, which is usually a positive when children need a bit more attention/input from adults. In fact less than 60 children in the whole school sounds like a huge positive. I'm guessing you have substantially less than 30 in a class?
It also sounds like they have a TA in the class which is positive because they are in short supply generally. The teachers addressing the behaviour of the few disruptive children in an ineffective way might be contributing, but there's no guarantee that this would be better at a different school. The only slight possibility is that a larger school might have had more experience dealing with disruptive behaviour and might have more practised/tried and tested proactive supports to pull out of a bag and try. For this reason, when you're looking at other schools I would actually be really upfront and explain the difficulties he is having in his smaller school, and see what they say about managing patterns of disruptive behaviour.
I understand the worry about the pattern where a child is struggling and the response is punitive and this leads to more disruption and more punishment which ultimately turns the child off to school/learning/authority in general, but (aside from the above scenario) switching schools is unlikely to stop this in its tracks. In fact, I think it could make things worse because essentially when you move schools there will be a period where they will say "Let's wait and see, he is probably settling in" and nothing will be done, meanwhile if he does have a skill defecit (e.g. communication, impulse control or emotional regulation) relative to his peers, the gap will be getting larger and he's not getting practice at the helpful skill you want him to develop. Most of the time, it's better to stick with the current setting and try to work with them to find the right approach for him. But also, do try to stay grounded in the present. Jumping ahead to an outcome which might not happen anyway is unlikely to help you address things effectively in this moment, because it's putting you into a fear mindset which means your way of responding is more likely to be emotional and reactive, rather than practical and proactive. I used to have CBT and my therapist would tell me "No fortune-telling!" which I try to remind myself any time I start doing this. You have a lot of various worries relating to Y6 and the transition to secondary school, but this is 6-7 years away - that's more than his entire life so far. Look at what is right and supportive for him now, that is a higher priority than perfectly lining up something where everything might have changed by then anyway. The best thing you can do to support him throughout school is to support with the challenges he's having now, that will give him the best foundation to deal with whatever the future holds.
If you're considering looking at private schools for secondary, then it's possible you have money to invest into supporting him. It might be worth looking at private OT (occupational therapy) and whether they can do an assessment and look at whether there are specific supports which would help him both in and outside of school, especially if it's possible for them to come into school and observe (I admit I have absolutely no idea whether this is a possibility.)
With the consequences discussion - you're right, don't punish him for something that happened weeks ago, but do talk to him about it, in a curious way. Ideally you want to get communication with the school to improve so that you're getting feedback much more regularly and it's more concrete. You hearing just a random stream of consciousness from the teacher about "He's done this, and that, and this, and that" and this is just negative all around, it's not constructive, it will make you feel that the teacher dislikes your child. Have the school actually asked you to do anything specific? Does your school have a SENCO and do you have a copy of their behaviour policy? It would likely be helpful to understand what the chain of escalation is, it sounds from your posts like "smaller" events are dealt with in class via loss of points/playtime but only the bigger events are relayed back to you, and at this point they are informing you that there is a general pattern of lower-level behaviour which is not getting reported at the time.
Consequences can work but they can also backfire massively if they are meaningless and arbitrary. If I thought ‘no pudding’ or ‘no TV’ would work, that would be great but they’ve had similar consequences at school (loss of playtime, loss of dojos, loss of privileges) and the behaviour continues. That’s because it is rooted in the dynamics of the social structure in the class. It would be nice to think ds would make sensible choices because he wants his ice cream, but it’s a bit more complex than that.
For sure. But the reason that people are asking you to try some simple consequences (and BTW it is a myth that arbitrary consequences are less effective, although it's true that distance from behaviour makes consequences less effective) is because children of this age generally find social approval/disapproval a strong motivator - in fact, this is possibly even driving some of the behaviour, because it's likely that he is seeking social approval from the boys he is showing off with and copying. It would be helpful, as a general measure, to show DS that you are displeased when his teacher has told you that he has been misbehaving in class as a general pattern, and many parents express this using some kind of unconnected loss of privilege which is why those examples are being given. Doing this a few times in the context of a relationship which is generally positive, supportive and accepting is not likely to cause damage to your relationship, and you never know, it might work. If it does work, then brilliant - problem solved, everything great. Surprising, but sometimes parenting is. (I have 3 DC. I have had to eat my words on all kinds of things I was absolutely 100% convinced of.) Where I would draw a line is that if it is not working, after a few times so you've given him a chance to test out this new response from you if it is new (even if this kind of thing does work, it is very unlikely to work after the very first time when it's not an established pattern) don't keep piling on punishments or disapproval, because that is where this kind of thing becomes counterproductive. Keeping it token, temporary, and where there is always another chance to try again and having that be the end of it is key in terms of it not becoming destructive IME, and research shows (counterintuitively) that a minor/token consequence is just as effective as a stronger/harsher/longer one, which is mainly because consequences themselves only have a very limited effect. You have to support the skill which is lagging or the positive, replacement behaviour to really have a lasting and concrete effect.
Have the school asked you to do anything specific? The other reason to try out giving some minor consequences even if you think it won't help is because it gives you something to come back to the school and say that you have tried, especially if it is not effective. Combined with the pattern of the sanctions in class not being effective to curb or reduce the behaviour, I would say that this probably warrants a meeting with more than just the class teacher, depending on school policy and which staff are available, it might be right to include the head, the SENCO if there is one, the TA who has experienced some of the behaviour etc. Anyone who works with children is familiar with the phrase "behaviour is communication" whether they believe in it or not. And without bringing up fortune telling about year 6, you can express concern that his behaviour is becoming a pattern which you do not want to become habitual, due to the effect on both your own son but also the rest of the class, and that he's experiencing repeated consequences which are not curbing the behaviour and your concern is that this could create a negative cycle now, and if consequences are not helping with the behaviour then he might need more support in a different way. Going into year 1, behaviour and academic expectations tend to increase, so I would definitely frame it as wanting to set him up for success for the next school year. See if the school have any suggestions, ideas or plans they can put in place to support his behaviour proactively, before he is in a position where they have to try and react to the behaviour.
You say that you don't experience behaviour problems at home, but you also say it is difficult to calm DS down when he gets into a "silly" loop, and it's probably (IME) this kind of state which is happening at school. It might be helpful if you can keep a sort of diary at home and try to notice if there is any kind of pattern leading up to the silly moods. For example, some children go into this mood when they are hungry, need the toilet or need to move (if movement is a need, curtailing playtime could be counterproductive). Some go into it in response to difficulty with the lesson, or difficulty with some aspect required by school work e.g. sustained attention to a non-preferred task, or following multi-step instructions. Some go into it in response to sensory overload or might be more prone to it when they are tired or feeling anxious because they don't know what is happening next. It is often a mixture of more than one difficulty. There are various things school might be able to offer as supports if between you you can notice any patterns.
Since you cannot avoid other children at school behaving poorly and you don't know what the root of their behaviour is - it could be something they are witnessing at home or anything really - do not focus on other children as the cause of the behaviour. Whether or not school think this, they will not be able to communicate much to you about other children. And I think the majority of 5 year olds would not be mature enough to ignore and walk away if they are being provoked. IME, the disruptive children tend to be drawn to each other because they lack skills in how to communicate and socialise effectively with the majority of the children, or because they tend to lack emotional regulation and impulse control so can be a bit "too much" for other children, but the other children with similar difficulties tend to accept this. The friendships can be fairly volatile with a lot of fighting and making up, and they can also have a lot of fun together. One conversation which really helped my eldest (who was not the one who struggled most) was about "classroom friends" who help you do good work and be sensible and calm, and "playground friends" who are great fun to run around with and let off steam but who distract each other.
One thing you might want to do is to try to strengthen other friendships outside of these more boisterous friends. Whether they are other children at school or children outside the school setting. This is a good approach (and would be useful for children without ADHD as well) - click on the download PDF link to read the article: https://chadd.org/attention-article/how-you-can-be-a-friendship-coach-for-your-child-with-adhd/