That is rarely true. If an organization ceases to exist but people still have the same goal then they create a new organization or act individually without an incorporated bureaucracy.
On the contrary, the existence of a mismanaged organization nominally dedicated to a given purpose often prevents its nominal goal from being achieved, because people assume giving time or money to that organization will be the best way to further the goal. Then the organization squanders them when those resources would otherwise have gone to some other organization or people with greater effectiveness towards the goal.
> On the contrary, the existence of a mismanaged organization nominally dedicated to a given purpose often prevents its nominal goal from being achieved
This gives rise to another type of person within an organisation. Someone opposed to the goals of the organisation, and who understands this all too well.
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.
* People who understand that the existence of the organization is necessary for the goals of the organization