The JCQ rules on who gets extra time are pretty strict, so I don't think there's any way someone in a private school could qualify, but they wouldn't have got it in a state school. Other access arrangements might be more likely to be "normal way of working", I suppose, in some private schools. But equally, I can imagine some private schools not really allowing things like rest breaks during extended work, so I would imagine overall, across both sectors it balances out?
I think the difference is having a school that can either screen all pupils, or parents who can pay for an EdPsych, rather than relying on a school who will have limited EdPsych hours and will prioritise those who are most obviously in need.
I do think at GCSE, in general, most students of average ability can finish the exams in the time allowed- but equally there are students who are in a bit of a grey area who are a little bit slower at writing, or need a little bit of extra thinking time but don't qualify for extra time, who might gain extra marks if the time allowed was a bit longer in some subjects (particularly subjects where an essay/extended response is needed).
At A-level, I've definitely known students who are pretty capable run out of time in exams- maybe not to answer every question, but certainly to go back and check their work, or maybe add a bit to extended response questions etc. Some students also just aren't great at exam technique, and if they get stuck on one question, they aren't so good at leaving it and moving on and coming back later, and then they may run out of time.
I've also known students who are really slow at one thing (e.g. graph drawing in science) run out of time- and again, that's not something that would qualify them for access arrangements, and you could say that's sort of the point! But they can do it, they're just not very quick at it!