There is a weekly 'good behaviour' reward at our school.
There are 38 weeks in the school year and 30 children in a class - if it were a case of 'everyone gets a turn' then indeed there is time, across the year, for everyone to get a turn. DS didn't get 'a turn' in the whole of Y1 nor in the whole of Y3 (once at the very end of the year in Y2), some children got it 3 or 4 times.
If it were a case of actual good behaviour being rewarded, then, by what the teachers and the headteacher tell me, DS should be getting the award every other week. But he didn't get it once (in two separate school years).
DS is not very competitive, but super anxious about behaviour. When we told the teacher that the reward system was making him even more anxious, the response was 'It isn't meant to make the children anxious, it is meant to encourage and reward good behaviour!'
Well the intention may be good, but that doesn't change the fact that for some children, the actual effect (vs. the intended effect) is negative - hightened anxiety, self-labelling as naughty, loss of confidence. But of course school holds on to its 'popular, successful system' and doesn't listen.
So we were in a similar position, OP, in that we had to deal with a behaviour management/reward system that our DS wasn't 'getting a turn' on and that was causing negative effects.
In our case we emphasized A LOT that we didn't care about the 'points' (in our case), but we DID care about his behaviour. We told DS that we wanted him to behave well wether this good behaviour was acknowledged with points or not. That we expected him to behave well because it is the right thing to do, not because there might be points to be earned.
We also had talks about how things didn't need to be rewarded when they came easy to you. That what some children were being rewarded for, were things that they really struggled with.
And we emphasized that his good behaviour was being noticed, and rewarded in its own way. For instance (we didn't say this to him but it is true): If there were situations where it was word against word, DS and a 'naughty' child, then generally, DS would be believed.
So we tried to find a balance where we didn't 'undermine' the system as a whole, but emphasized that for him, it was his good behaviour that we cared about, rather than any certificates or awards.
But personally, I hated it - you should have seen how incredibly anxious it made our DS. To a point that is unhealthy. It took a lot of work by us to ameliorate the situation, with zero help from the school. The system, meant to 'encourage good behaviour', for DS caused rather significant mental health issues. I am not a fan, and get very cross whenever I hear 'ahhh poor poppet, but it's not meant to make you anxious' (or similar).