Having experienced a grammar school and a selective independent with my two, I'm sure there's no difference in teaching/teacher quality compared to comprehensives. What I have noticed is there's much more stability in terms of teaching staff and far fewer supply teachers. With both schools, there's usually a handful of teachers leaving each year, often less than 5.
There's also the general environment of aiming high (more so with girls than boys!) which can be helpful for the naturally lazy, but it's also a bit stressful. With our daughter, we've needed to constantly remind her that she's with the top 5-10% of the ability range, so struggling in that environment from time to time isn't really struggling, if you see what I mean. She's found the other girls incredibly supportive though.
I think, given the general issues in education - underfunding etc- on top of covid, that a grammar is a generally safe bet, educationally. That's one of the reasons dd is moving to a grammar school for sixth form and not a college or comprehensive.
We didn't get tutors for either of ours (and we're in a super selective area) but we did do a bit of work with them at home - about 3 months worth with ds, less for dd. When they got in, they were able to cope with the pace.
We didn't let them choose whether to take the exams. They're too young at 10 to understand the options and the implications of each.