I would just like to weigh in with my experiences with my two foster children. In our, very select, personal experience, offering rewards to 'naughty' children, when their behaviour is still often poor, doesn't always work.
My youngest came to me in Reception. He wouldn't do what he was told at school but most of the time it was free play so he coped ok because he was mainly doing what he wanted anyway.
In Year 1 he had a horrendous year. The trouble was, the teacher didn't tell us how bad his behaviour was getting until about the February so we assumed he was managing quite well and his behaviour was improving; as it was at home. But actually he was running riot at school. It pretty much boiled down to the fact that his teacher couldn't get him to do work so he didn't do the work. She gave him an inch and he took a mile. He was climbing on tables, chucking chairs, shouting at adults, hurting children, all sorts. They tried changing his class for the last half term and that had no effect. They couldn't get him to do what they wanted so he just did his own thing. Yet at the same time he was bringing home prizes and things - he once won a competition for best picture which was simply scribble (and we did lots of crafty things at home so he was capable of more if he'd tried). I'm sure other children's must have been far better but I bet the teacher couldn't face a tantrum from him. Which I do understand but by giving in to him all the time he was getting worse and worse.
At home he was completely different. He soon learned that chucking stuff around etc didn't give him his own way - he still wasn't allowed to do whatever it was he'd been told he couldn't do, so the tantrums soon stopped. He was still no angel but we didn't expect that, he'd obviously been through a lot of trauma, but he became manageable. We had clear expectations and consequences and we did use reward charts in the first year or so. At school he had a reward chart but he didn't care whether he got the stickers on there or not. Basically he didn't respect the teacher and didn't feel that they were in charge.
He did really well in Year 2, which was at a new school. Suddenly he wanted to do well for his teacher. My biggest issue then was the school was suddenly expecting him to meet the normal Year 2 academic targets when he'd come to me at the end of Reception knowing nothing so I didn't expect him to have caught up yet, and I wasn't bothered that he hadn't - I just wanted him to behave in school! And, with a couple of minor blips, he did. What really helped him was having a strong teacher who gave clear expectations and consequences. He's the kind of child who can't take being praised constantly with no consequence for bad behaviour, because he will push and push at the boundaries. But if he knows the boundaries are secure and firm he will (usually!) stick to them.
What I found a bit difficult to deal with was that for one misdemeanour he'd had, which by then was out of character for him, he wasn't allowed the end of year treat. I thought it a little harsh but he understood what he'd done and exactly why he couldn't have the treat so fair enough. But his brother, in Year 3, was allowed the treat. Yet his brother's teacher had had to speak to me most days for a couple of weeks about his behaviour! So that was difficult to explain - why his brother could still have the treat even though his behaviour had been much worse! That was really due to the teacher being lovely, but a bit soft, and letting him have it anyway. Whereas the younger one's teacher was holding him to the standards of the rest of the class, which I was pleased about really because it showed how far he had come - if he had stayed at the old school I honestly think he would have been excluded by now.
He's now in Year 3 and his class teacher has no real problems with him. Yes he's bossy and opinionated and he fidgets too much and still hasn't caught up academically, but he manages at school behaviour wise and doesn't need any special reward system other than what the school uses as a whole school system, so I call that a real success!
I honestly think that if he was allowed to be naughty but still get treats his behaviour would just keep escalating. It might work for some children but it doesn't work for him. He needs to know clear rules and he needs to know he can't break them.