"Mainstream" parenting, including advice given on courses run by Sure Start centres and early intervention workers, such as Triple P and 1, 2, 3 Magic, uses things like reward charts and naughty steps and grow clocks and sleep training to disciple/control/guide young kids. I choose not to, and the term most closely matching how I choose to parent is Gentle Parenting.
The thing is though, that there is a lot more overlap with those programs and your definition of gentle parenting than you would expect if you go by purely GP resources (which tend to be quite anti "anything mainstream", which is a weird distinction).
I used to think this too, but then I actually learned a bit about modern behaviourism. I thought that behaviourism = associate unwanted behaviour with something unpleasant (punishment, disapproval, shame) so that the child/person/animal/etc doesn't want to do it, and associate wanted behaviour with something desirable (approval, praise, reward) so that they do have reason to want to do it, but it turns out that is just outdated. Like really outdated, as in, nobody should have been teaching that for at least the last 30 years, and anyone who has been is just wrong/out of date.
Modern behaviourism says that for every unwanted behaviour, it is possible to identify what they call a "positive opposite" behaviour, which is defined as something that replaces or is incompatible with the behaviour that you do not like. The focus is then on coaxing that behaviour, through a combination of using the right wording, playful invitations, descriptive praise, recognising small improvements/steps towards the wanted behaviour, helping the child build skills and practice related skills to that behaviour, making sure that YOU as the adult role model, are modelling that behaviour (and especially not the behaviour you don't want). And yes, rewards. Sometimes, for a limited time for specific goals towards a wanted behaviour. And with all this, in theory you never need to even do anything towards the unwanted behaviour, because it will disappear all on its own through you doing this.
This is also a really huge concept in gentle parenting. As in, one of the main concepts. So I think some people claiming gentle parenting is so different to "these mainstream programmes" would be really surprised to note that there is such an overlap here.
Secondly, most of these positive programs have a huge emphasis on relationship and connection with the child, which again is a large part of gentle parenting, along with predictable routines/rhythms and stability. The parent being a calm influence rather than chaotic.
In fact, the only real difference that I can see between positive parenting/modern behaviourism and gentle parenting is that gentle parenting explicitly puts emphasis on the role of emotion in children's behaviour and encourages the validation of emotions and practices of co-regulation, and explicitly says do not worry that comforting your child is going to reward their unwanted behaviour, particularly if that behaviour is actually a physical manifestation of an overwhelming emotion. I can't find this in any of the positive parenting/behaviourist/evidence based programs, and I don't know whether that is because it has been studied and found to be ineffective, or whether it is that there is no evidence supporting it, or whether it is simply considered to be irrelevant because behaviourism is concerned only with behaviours, so feelings do not matter. (The behaviour might be crying, or jumping up and down, or talking at a fast pace). And actually, behaviourism (even modern) DOES actually say that giving a hug/attention/trying to talk to a child while they are engaging in the behaviour that you do not want is incorrect, positive reinforcement and will unwittingly reinforce the behaviour that you do not want.
I do think this is a weakness of behaviourism. I find it a bit cold. I think that it is often worth looking to the root of behaviours instead of simply responding to what is happening on the surface - this is also really key in attachment research - children must have attachment figure(s) who listen to them and see them as an individual rather than a collection of behaviours. I think (especially young) children CAN be overwhelmed by their emotions and they are not necessarily engaging in consciously manipulative behaviour in those moments, so I do not think that sympathy, trying to talk to them etc is reinforcing the behaviour in the way that behaviourism as a theory suggests that it is, and I actually think that denying them comfort or support with their feelings, because they expressed those feelings in a way that you did not like, because they are little, is cruel. (But I'm not a behaviourist; maybe I have misunderstood this).
To the question of punishment this is interesting, modern behaviourism/positive parenting does not actually need it, in theory, and emphasises that it should be minimised and only used "when absolutely necessary". But the problem that a lot of parenting programs face is that they are written for parents who are starting from a point of being much too punitive, possibly even abusive - this is especially true for the ones run/funded by government organisations, they are one aspect of social care and trying to improve circumstances for children in homes where it's not quite bad enough to remove them, or as a last ditch attempt before children are removed since it is better if children can stay with their parents if it is safe to do so. So if parents are ALREADY relying a lot on punishment, or threats, or violence even, it's ineffective to just say well don't do that then. Like when you're dealing with child behaviour, you need a positive opposite to encourage in the parents. Otherwise, they tend to fall back on hitting, screaming, threatening etc. And that is where generally these programs use time out, because it's extremely minor - no child is traumatised by being told calmly to sit on a chair or a spot or a step. It is also simple and can be implemented anywhere so a parent is not stuck without access to their coping mechanism. It's non violent and it is formulaic. The other options offered (given generally as a fallback if time out is proving conflict-inducing) are again mild and do not require cooperation from the child so they remove conflict from the moment - removal of privileges, or removal of points from a points/token system (which is otherwise used as a purely positive motivational tool). But the point is very much not to use these as a primary method, because the primary method of behaviour change is positive reinforcement, and building of relationship, and predictable routines, and the way that you ask for behaviour, and having reasonable expectations (with the steps from the unwanted to wanted behaviour) and so on.
Being formulaic is something that gentle parents do not like very much. So you get a weird dance in a lot of gentle parenting spaces where "natural, or logical consequences" are okay, but "generic punishments" are totally out and I used to think that this made sense, but I now think this is a total fallacy and a guilt assauger. Because what you actually see is parents twisting absolutely anything in a knot to convince themselves that it is a "natural consequence" when really they want to associate an unwanted behaviour with an unpleasant outcome for the child, ie, they want to punish.
That is natural because it seems logical. But it is not evidence based and it is not an effective way to change behaviour. And if you really want to be nitpicky, it's not in any way gentle! What would be more gentle is recognising that as parents, sometimes it's really hard not to react to something punitively, therefore I should have something token and harmless and basically neutral in my pocket to pull out at times when I want to punish but I know that I should not. Know that that does not actually change the behaviour, but it stops me from doing something worse in the moment.