100% this. I have also seen families taking DC out of private school for A-Levels so that they can have a competitive advantage to get into top Unis.
They are hugely misguided.
I've worked in student recruitment and widening participation, and had plenty of contact with admissions.
Universities can see what schools applicants went to from 11 onwards, so they're not fooling anyone.
Grades are typically higher in private sixth forms; those higher grades will be the most important part of their application.
UCAS advice is often better in private sixth forms; having visited more schools than I care to remember in both sectors, there often is a huge difference. Some of the most fulfilling memories I have from my time in the sector were filling in the gaps in the advice they'd received; I'll always remember the girl who told me she was torn between a degree in MFL and a journalism career; no one had told her that an MFL degree could be great preparation for a journalism career. The look on her face was priceless. I digress, however...
WP schemes have much wider criteria than going to a state school. Often they look at free school meals, having been in care, caring responsibilities, being at an underperforming school, parents didn't go to university etc. Indeed it's possible in some circumstances to attend a private school and get onto a WP scheme such as https://www.leeds.ac.uk/access-to-leeds/doc/am-i-eligible
It is unusual for the deciding factor between two applicants to be the school they attended.
If I were short on funds but trying to maximise university entrance chances for my DC, I'd send them to a good state school for GCSEs and private school for A Levels. Sixth form does seem to be where you get more bang for your buck in that regard.