I have been a member of both private clubs and also clubs that were part of public pool room. As far as playing goes, I very much prefer private club because playing time is cheaper in the long run and you don't have to put up with a lot of crap that goes on in public rooms sometimes.
However, for private club to work at all, even if you can cover all the expenses comfortably with just a small monthly membership fee per person, you still need 100% of integrity from all people involved, i.e. owners, founders, members...etc. This is absolutely vital. If there's any animosity at all amongst the players or the owner is using money in a non transparent way (to put it politically correct:grin

then the club does not have much of a future. That's the story of my club unfortunately.
I play in a private club, a very nice one indeed, at least it used to be, but it is all going to hell. The founder/owner did not pay for the rent and electricity for several months back despite being paid member fees on time. He did not appear in the club for three months now!!! He owes a lot of money to a lot of people, including some to actual club members, appears to be broke even though he claimed to be a millionaire (I was naive enough to buy that stupid story). Nobody pays anything to him anymore, there's no confidence anymore. The funniest thing is that the tables that we play on are his on paper, but he tricked the real owner and paid only about half the price they are worth (of course using borrowed money from someone else). So the real owner of the tables has been chasing our "president" for money for more than a year now. But there's nothing he can do as the real owner is from another country. Ouch! This is just a very short version of the story...