In England, many schools are not run by the LA, but by academies/multi-academy trusts. These are generally not for profit organisations, but they take on the management of the school budgets and receive funding directly from central government. They are responsible for providing all services like HR, payroll etc to their staff as well.
Schools are not meant to have a deficit budget, so if the MAT runs out of reserves, they have to make do with what is available to the school. In general, schools aren't meant to pay staff salaries via parental donations etc either.
When people talk about schools going bust, they really mean the MAT has run out of money. In reality, what that means is the school will continue to run, but it may have no funds to e.g. pay supply staff, so if lots of teachers are unwell one day, the school would have to close (as we saw during covid).
That said, I do think things will get tight, even for LA schools in England, because as has been said, local authorities have no money either- so they won't be able to bail out schools with ever increasing costs.