What happens to the original Clubs (over generations)?

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.

When I deleted all the premade Sims for my Empty World save (re-filling the game with my own Sims and houses mostly), all the clubs were deleted, but the game developed new ones including names. My favorite is the fishing club called One For The Halibut. :D

I use clubs mainly to gather Sim easily for social events. I have my Happy Fun Club for my child Sim, though that one will soon be disbanded because my Sim and her friends in the club will be aging up to Teen, at which point I'll start a new club so they can go out bowling, nightclubbing and karaoke singing. Often as family members age and move out I'll make a club for the YA+ siblings so they can gather to go out to dinner or for holiday dinners at home.(Can get tricky when you've got a large family so I usually limit it to just the siblings, no spouses/partners.)

The only Maxis club I use occasionally is the Powerhaus club for any sim doing an athlete career, otherwise I don't use them.

My favourite self made club I use is the Science club, a park lot with chess tables, Observatory, Microscope etc., with settings to use those items and friendly socials. :smile:

I thought once of starting clubs where sims made items on the wood working table or baked goods or any other items they could make and then sell the things in a retail store, bakery or other business - but actually never tried doing it as I never really developed a big interest in the clubs to try my ideas. I also was not so sure it would work - that the club members would pocket what ever they made. LOL. So just never ended up messing with any clubs.