Saintly, I'm not trying to discredit research by professionals, but I think the main point is that these things are not implemented the way they are intended to be. In fact, I know they aren't. At least they aren't in the outstanding school I work at, which is in my opinion staffed with very competent teachers who genuinely care a lot about all the children in the school. That's not because they don't want to get the best out of the system and out of the child, it's simply because they don't have the time and the resources and the support.
It's great to say that we should identify why the behaviour is happening. But that isn't always that easy in reality, especially with a child whose parents don't speak very good English so can't give any insight from their point of view. Then there are the children who have had multiple observations done on them and every display of aggressive behaviour has been documented, including what happened just before the incident. Sometimes there are no triggers. Or the trigger is something as simple as a child not getting their own way, which in a classroom is something that cannot be avoided. Sometimes you can explain why the behaviour displayed is bad, but there is nothing that can be done about it anyway, especially with unsupportive parents. I mean things like having to deal with a sibling who has severe SEN, or a parent who has health problems. When we can explain why the behaviour is bad, we need to be able to deal with that, but we just don't have enough access to school counsellors or child psychologists or time to thoroughly talk through a child's worries. Other times we can see that the child is probably frustrated just because they can't keep up with the work, so of course they will get some extra help, but unless the parents are prepared to put in the time and effort we can't do as much as is needed because to do so would mean holding back other children.
It is so frustrating when things like that happen, because we want to help, we want to support the child, but there just isn't the time to do that when there are 29 other children who deserve their fair share of the attention and teaching time as well.
What you say about staff doing a functional analysis makes a lot of sense, but my worry is that all too often this doesn't happen. Or if it does it's not done properly. Then you end up with a child being put on report and being given stickers and prizes as described in the OP without effective strategies surrounding it, and that not only doesn't help the child in question get the most out of it, but it also sends a confusing and sometimes damaging message to other children.
I disagree with you that rewards being used inappropriately is not a reason not to use them. I think it's a very good reason not to use them, for the benefit of all the children concerned. If it can't be done properly, I don't think it should be done at all.