That's correct, there is no profit, everything is put back into the school, so the budget at the end of the year is 0.
I think it's debatable whether they are to the public good or not. In some areas I'm sure they are - taking pressure off school places, leaving more money to spend on state schools (as parents who send their child to a private school still contribute through their taxes to the state schools). Some will share facilities, extend Young Enterprise schemes etc.
In our area they are a total nightmare. There are a HUGE number of private schools, and virtually no good state secondary schools. The council seem to rely on the brighter children going private, and the state schools take the rest. Our results are shockingly bad - 3rd lowest in the country. In the whole of our city, only something like 35% got 5 or more GCSEs. For people like us who probably can't afford private schools for 3, it is a nightmare. If the children at the private schools were in the state sector, the city would HAVE to accommodate them.
They built a new school a couple of years back, which should be a good one, it is in a nice catchment area (we are not in it alas). However, on the gossip grapevine I have heard that lots of the people going there are actually people whose parents have sold an expensive house in London, moved to where we are (120 miles away), bought an expensive house right next to the school so they will commute back to London and their children go to that school. So not solving our schooling crisis at all.
Who said money can't buy a good state education?