OK so you say that you thought perhaps speech was an issue before, but it didn't get better. Has his speech improved? You say no concerns with speech, but he didn't tell the boy to get down off the climbing frame, or invite him to play, or talk to him at all - he went straight to kicking?
I don't think that nursery saying there are no special needs means that it's definitely just the way he is or parenting. It's like you're seeing this as a black and white issue but it is not. The problem is that at three there is such a range in development and behaviour.
So you will have children who are fine at nursery, and respond well to basically any reasonable discipline technique and are naturally eager to please and will adopt prosocial behaviours by coping other children and adult role models, listening to messages given by teachers, parents, books, TV etc.
Then a little bit further down the scale, you'll have children who are a bit more challenging, maybe not in all settings, they will respond to discipline and they get there with the good behaviour, it just takes a bit of time and consistency.
Then a little bit further down the scale, you'll have children who have more persistent behavioural problems, they don't naturally pick up the social behaviours, they don't necessarily have any SEN or disorders, but they do need a bit more targeted behaviour management specifically to teach them the correct behaviours or work on the skills they are struggling with or just scaffold them a bit until their development catches up. We can't all be good at everything, and while some children will be really social but struggle with maths for example, there will be children who struggle with social skills, or emotions, or impulse control. Not to the level of it being anything diagnosable. Just needing a little bit of extra support with learning that.
Then you'll have the kids that do have some kind of SEN but it's so borderline it's hard to detect, their behaviour/milestones aren't significantly off track but they just seem to be that bit more difficult a lot of the time. IME, nurseries and schools typically don't alert to these cases because it requires a more skilled, experienced eye.
The cases of SEN that are so clear that nurseries pick up on them are quite far at the severe end of the scale IME. These children stand out quite clearly from other children their age. I think this is probably why those "borderline" cases don't alert nurseries, because they are looking for a presentation like this, whereas they may miss a more subtle one.
For example, I live in another European country where there are regular developmental check ups. Nursery never picked up on my middle child needing speech support but his paediatrician did. In the UK unfortunately you don't get this automatic yearly check up so it's up to you to ask your health visitor if you have any concerns about your child. Honestly, I probably wouldn't have thought to. In fact, my eldest child's speech was similar and when we lived in the UK I did not get him looked at because I did not realise that his speech would be concerning.
Which is all to say - I don't think that you necessarily need to be thinking that your DS has SEN to look for support or try a more focused behaviour coaching type approach. I can also understand if you're wary of your child being labelled, but to me it does make sense to rule out really easy things like a speech delay, hearing issue, stomach pain, glue ear etc.