My kids' last school was fantastic about this sort of thing. Main parts moved around the class. Assemblies and plays were designed to spread participation, not concentrate it on a few people. And they worked really hard to make sure all the children, even the shy ones had a positive experience speaking in front of an audience. They put a lot of time and effort into it and it really seemed to pay off. At the end of every half term there were two children chosen from each class to go into the "Head's Book" (on display in reception). One child was voted on by the class and one chosen by the teachers, no child could go in the book more than once a year. Voted children did tend to being the popular ones, but the teachers balanced that out with their picks. Chats with parents further up the school (and older children) confirmed that everyone got in the book a couple of times before they left.
There was a star of the week, and deputy head teacher rewards for special effort and behaviour. These were used to reward children for stretching themselves - whether they were academic stars (like one of my children) or "invisible middles" (like another), or children with special challenges (like one of my kids' best friends).
The only area where this did not happen was sport after the hiring of a PE specialist. He focused very much on the "stars" and on traditional competitive boys sports at the expense of others. This was a real shame as my kids - both in the 5th centile for size and not particularly physically adept - were beginning to get into sport before he came along.
The teachers of the classes my kids were in were particularly good at thinking about the activities it did in terms of how they developed the class, rather than simply thinking - oh, now it's school play time, what's the easiest way to do this. Which I think is what happened at school when I used to go. My kids class teachers were phenomenal though. I was almost always highly impressed with their skill and dedication.
When I was in school it was very different. In primary school plays were about impressing parents, so same confident children starred every year (of which I was one). Prizes were for attainment not effort. So older children in year almost always got them (again, I was one). At middle school things changed a little, plays, sport teams etc. were mainly lunch time and after school activities. Everyone could participate if they wanted but didn't have to, so self selection was a big part of it. Confidence was not enough for a starring part in the school plays, our drama teacher favoured stage school kids and it was very much a case of show casing a few kids rather than developing something they could all have big parts in, but they were spectacular productions and that in itself was quite an experience to be a part of, even dressed up as a non-speaking six of hearts (my biggest role). PE was worse - the good people got chosen to captain teams and the rest of the classes were "picked" by the captains, one by one. So the better and more popular kids picked early week after week and the not so good ones left till the end every week. Horrendous way to run a lesson. School teams were Open to everyone, butit was the same people captained or played the majority of the game every time with others slotted in for ten minutes here and there. Art was much bigger at middle school and there were a lot of competitions, invariably won by the same few kids. Art lessons were more inclusive but attention and time definitely went on the ones the teacher thought were talented. Music lessons were just (poor) childcare with no consideration or time given to children who weren't already musically accomplished. Some of us were even told to mime rather than coached on how to do better, when we did singing. I excelled at sport, so my lack of talent in other areas didn't bother me much, though my appalling musical education has haunted me ever since in terms of confidence to, for instance, sing happy birthday at parties.
I have some sympathy for the action of teachers giving up their free not wanting to have children who don't want to be there forced on them. But most clubs are voluntary so you only have the enthusiastic. A teacher who only wants the talented rather than the enthusiastic, strikes me as being in the wrong job and primary and secondary school level.
I think there is a lot to be said for having some activities that are engaged in the pursuit of excellence for those that are talented/work hard. But schools must not simply give up on those who aren't excellent in order to free up resources for those that are. I'm really glad to say there are a lot fo teachers out there who don't. The teachers my kids have had at school have been heads and shoulders above the ones I had at their age in terms of skill and dedication in teaching all children. It's such a shame government policy seems to constantly undermine them.