I'm sorry OP, it sounds like you're having a really tough time.
No one on this thread knows if he has additional needs (and 5 is still young enough where behaviour might point towards SEN, or it might be within the 'normal' range of behaviour). You're right though, SEN or not the challenges remain.
It sounds like you're a really good parent. Some children, for a whole host of reasons, are just more challenging to parent.
If you can, try to separate your son from his behaviour and address it almost scientifically. I'm sure you've already tried loads of approaches, but try making a list of ideas then try them for a couple of weeks, making notes on his reactions.
My three year old reacts similarly to your son if he feels 'told off', recently I have tried this really cringy approach: "Oh, you're such a kind boy but that wasn't a nice thing to do/say. Why don't you try again, I know you can do it!". It feels ridiculous when I want to tell him off but it works so well for him! Before I tried this I had used consequences, reward charts, explaining why it wasn't nice etc. Everything turned into a meltdown.
My 5 year old doesn't respond to this approach at all, but will usually stop if I give him a warning about leaving or a time out... they are all different.
You could also try speaking to his teacher. If you find out the behaviour process they use at school and this works for him, emulating it at home might help with consistency. Or they might be able to point you in the direction of some help. Barnardo's sometimes run drop in sessions for different issues. It can be aimed towards SEN children but rarely requires a diagnosis and often the advice works regardless of SEN.
Please try not to compare him or yourself to other children/parents. It's easy to assume parents with easy children are doing something 'better', but often it's the luck of the draw (my eldest is the one with diagnosed SEN, but if you judged me on his behaviour you'd think I was a much better parent than if you judged me on his younger brother!).