Your assumption is correct.
Once the last member dies or becomes disqualified based on requirements, the club ends.
The game will generate new clubs with different rules to keep a minimum number of clubs active.
If you were to use MC Command Center however, you could keep clubs active and alive if you use the settings to recruit eligible club members.
As for families in clubs, it depends on what type of family and where the club gathering is, but I often let the others tag along and do their own things while one of them is in the club meeting. Multi purpose community lots might be a good idea in that respect.
Personally, I'm not much of a fan of the Maxis made clubs, but I do think the Knights of the Hedge are pretty hilarious.
