Yes, I would seriously consider that - although I would want to think carefully about the division of group and individual terms, to ensure that individual renters did not get stiffed on things like maintenance and so forth (and likewise for the group to be legally insulated from the defaults of individuals who fall behind on the rent).
I've been home-hunting in SF for going on 2 years now, and it's been a miserable experience. The rental market is not much better; a lot of the housing stock in SF is really shitty, and prices are being kept artificially high by irresponsible lenders and borrowers. This market is ripe for disruption and there is definitely room for alternative business models. What you're proposing sounds somewhat similar to housing associations I've seen in some European countries.