Yes, I wasn't giving the link to suggest that he has ADHD, just that it's a useful approach socially for children who struggle with this specific aspect of getting overexcited and hyped up and then missing social cues from others. That is common in ADHD, and it's published in an ADHD magazine which is why that's the title, but the intervention is essentially a version of social skills training which is designed for use within the environment the child is experiencing the difficulty. The problem with social skills training usually is that when you're 5 and get overexcited by your best very very fun friend doing something crazy, there's no way you're going to think back to the therapist's office and the role play you did with a doll or the book you read about "hands are not for hitting". You're just caught up in the moment. So they need the training in the moment because this is what can help rewrite patterns of behaviour. But standard, reactive in-the-moment discipline strategies are unlikely to work because the skills they need come several steps before the behaviour which gets caught in a reactive discipline approach. That behaviour is "late" to quote Ross Greene. It's more of an outcome of the way they are behaving early on, which is not necessarily picked up on because most children can behave exactly the same earlier on in the process but will instinctively stop at some point where it feels too much for them. For children who are getting so overexcited that they miss those cues, or where the inhibitory control ("brakes") is not well developed, whether this is related to ADHD or just difficulties in that specific area, they need more support with this part of the interaction, where the cascade of behaviours and reactions starts which results in a behaviour that is going to get them hurt or told off.
Since you had a reassuring conversation with the headteacher, and if you/the teachers are all in agreement that the combination of the three of them is the catalyst, I don't think it sounds like they are labelling him a problem child. It sounds to me like they've been trying to address the behaviour in class and are now at the point where they need to step up that (hopefully to offer some rounded support) and so they are cluing you in, which is exactly what they should be doing, but might come as a bit of a shock to you if they haven't mentioned anything before and it seems like suddenly there are all these issues and it's an emergency and needs handling straight away - that's a completely understandable reaction, but if you can step back and look at the bigger picture, it doesn't sound like it's at an emergency level to me. It just sounds like they want you to be aware, which is great.
I would definitely keep pursuing relationships with other children in the class and outside of school, and try to ensure these interactions are almost always positive, because it can be a very useful parallel to draw with DS when you say something like DS, when you play with Jake he's always nice and you have fun, but sometimes when you play with Simon he is a bit mean to you, have you noticed that? He probably won't get it yet (and don't over-egg it or he'll feel he has to defend "Simon" to you) but I bet as he gets closer to six or seven, you will see him start to reflect on the fact that his friendships with less turbulent children tend to be more enjoyable and he gets into trouble less. And although I also don't especially like consequences (esp arbitrary ones) as a method of "teaching" and I don't think they really teach anything or lead to behaviour change on their own, they can be quite useful in terms of helping children join the dots on something which might otherwise be a bit far past their understanding, because he will probably spot that when he plays with child X he ends up with consequences whereas when he plays with child Y he doesn't. It's a bit cruel to be kind, but it can help. This is also why to make them minor (or reverse them e.g. offer a sticker/point for a period of time without certain behaviour) because it helps when you're frustrated and feel he's not developmentally able to distance himself from the more boisterous children, but there isn't any immediate natural consequence to going OTT with them, in fact it's actually a lot of fun. Sometimes the job of arbitrary consequences is to be a proxy for a natural consequence which would be too severe or too far in the future (such as every child in his class thinking he's naughty and to be avoided). We can use adult foresight to see where that behaviour pattern is leading. If you're expecting a consequence to be scary enough that it stops his behaviour in the moment without an adult interrupting the behaviour and reminding him, then no, that is not likely to happen (but this is good - you don't actually want consequences that scary!!) it's more that it builds a picture over time and they get a bit fed up of being the one who loses out on the sticker or has to go inside or whatever it is, and they see that as a consistent pattern over time, then it can get them to see the perspective of OK, maybe I do want to change how this goes. The school consequences might be enough for this BTW - I think you'll get a sense of whether they are or not.