I think it's quite misleading to talk about "private schools" versus "state schools," It makes more sense to talk about "selective schools" versus "non-selective schools."
The fact is, private schools will always achieve higher numbers if top grades than state and this is because most private schools are selective to some extent - and many are super-selective. Just like grammar schools have always achieved more top grades than comprehensives. Similarly, you can't really talk about Brampton Manor in the same category as your average "state school." It's not comparing like with like.
It may well be that a lot of private schools inflated grades more than state schools last year. But this was not the case with the super-selective in London which are, between them, where a high proportion of Oxbridge entrants come from. You can see that the proportions of A star / A grades awarded this year are consistent with last year and many years before. If a school like Westminster or St Paul's is getting 96% A star /A grades year on year, there was not much room (or need) for grade inflation through TAGS.
If anything, super-selectives were more cautious with the TAGS because these schools all watch each other very closely (a swing of less than 1% in grades can mean a school is 10th in the league tables, rather than first or second - margins are that tight).
For instance, my DS is was at a London super-selective. They did exams as if A-levels were happening and results were 100% based on performance in these exams. No mitigations. But , even in certain MFL where the whole class achieved over 95% in all papers, they still didn't want to give them all an A star because they knew this would attract an investigation by the board and possibly negative press. My DC was one of these to be awarded an A, having achieved 96% overall and no less than 94% in any paper for that subject (oral, listening, written, etc). He appealed to the board and was awarded the A star that the school should have given him in the first place. And he wasn't the only one. So if anything, I think certain super-selective schools were over-cautious with TAGS.
Compare that to a local sixth form college where they did a variety of mini "tests" to inform TAGS. The students were laughing to my DC because the school didn't even bother to take phones off the students (!) and they could all confer (copy each other) in the tests and Google the answers!