'I would like to take him horse riding but am a bit worried he'd frighten the horses, flapping and jumping. Is it that sort of thing?'
Yes. It is exactly that. If you feel that he would benefit from horse-interaction and the stimming is getting in the way, then he might need to learn how to control the stimming for just long enough to see the benefit of 'not stimming'.
My son attends a choir for an hour. It is a good choir and difficult for kids to get in. Standing still is VERY hard for him but a part of the whole choir 'thing'. A good choir is considered so from the discipline of standing uniformly. Standing still is even MORE difficult when it comes to a performance where he gets both anxious and excited.
We had to do a LOT of work on practising the 'non-flapping' thing. But it wasn't me saying to him 'right you're going to choir, don't flap or you'll be in big trouble!'. That would give him mental health problems.
We practised short chunks of 'not-flapping' building up. We talked about the opportunities the choir might give him that are close to his own ambitions and goals. We schedule flapping opportunities before and after choir practise. And a safe space whilst waiting for his 'slot' at performances. His choir leader supported this and helped.
For a young child it might not be as easy to have discussions about it so you might have to motivate him other ways. A BIG minute timer that he can watch and after successfully not stimming for a minute he gets an ice-lolly. You can call it the 'minute game'. You are not trying to eliminate the flapping out of his life, but teach him that sometimes, if there is something better to do, he can stop, just for a minute to see if he agrees.
This time can be extended as he grows. In school it might be necessary to stop stimming just long enough to hear the teachers instructions, but he's 5 and all children that age are impulsive and want to do what they want to do. It's unfair to single him out just because the teacher doesn't like it or it doesn't fit into their EYFS tick box.