The grammar I know of get less govt funding not more. BUT, they have a very active alumni group and parental involvement organising various fund raising events, such an annual gala, running the school "shop" selling uniform, stationery, gifts, etc, social events, car boot sale and table top sales, etc and an active charity subsidiary. By keeping in contact with alumni, they do get a lot of regular donations once the leavers have got established in their new careers and earning. There are regularly quarterly newsletters, plus email marketing lists, plus active social media etc., all to keep the awareness of the school among parents and alumni. All of that is done entirely by volunteers, no involvement from school admin/teachers at all. They also had big fund raising campaigns for buying "big" things, like new sports equipment, repairing the leaky roof in the library, a 3d modelling machine, etc. There was an entire ethos of keeping parents closely involved in all aspects of the school and keeping contact with leavers. Lots of leavers go back occasionally to do career talks etc.
By contrast, I never heard a thing from the crap comp I went to, the moment I left I was forgotten and I forget the school. Whilst I was there, there was zilch in the way of parental engagement other than the usual rushed parents evening - no PTA or anything like that, no "events" where parents were invited, no fund raising. It was as if the school deliberately avoided any contact with parents or leavers, so they missed out on huge potential help, support, finances, etc.
I think that's the difference really. Grammars and private schools really seem to want to involve parents and leavers to create more of a long term community that they can tap into in so many ways.