The most important factor in getting an interview for most medical schools is the applicants score in the entrance tests ie BMAT and UKCAT, as most applicants have good grades / predictions.
These tests are identical for all candidates regardless of background.
Most University widening access policies focus on schools / home addresses in the bottom 20% and other under represented groups, such as those who are care experienced. It’s usually not as simple as state vs. private .
Such pupils might be selected for interview at a slightly lower mark than those from schools who sent more pupils to university. But it’s only a small difference, usually 10%.
So moving from a public school to a grammar for A levels will not increase your child’s chances significantly . You’d need to move your child to a very poor school for at least 4 years AND make sure they got a good score in the relevant test.
It’s really not simple to game the system. University admissions officers are not that stupid.