In the UK we have a variety of arrangements for schools. Some are local authority managed, some are 'academy status' which means that they are self managed but often with a cluster of schools sharing a management layer to save money. There are also 'free schools' which are community run with often an 'alternative' ethos. And there are religious schools, run by churches (and other religious organisations). All of those are state funded using a funding formula, and they have to teach the national curriculum, and are subject to inspections. Academy status schools used to get a bit extra but not any more, they can however employ staff who are not qualified teachers (Qualified Teacher Status is a defined set of training and experience requirements).

There are also private schools (some famously called public schools like Eaton or Harrow, but most actually just private companies often with charitable status).

Schools are usually fairly small organisations and generally the management have risen through the ranks as teachers, year heads, and so on. It isn't a sector in which fortunes are made.

So, yes, I think a range of funding and organisational models are possible. But note the role of regulation (direct inspection of what happens in classrooms on a regular basis without much in the way of warning).

You seem to be questioning the possibility of private schools existing when they obviously do. Moreover, you could have publicly funded education without having a state-operated school bureaucracy, or without that bureaucracy having a monopoly on the funding.