Such villages exist, but honestly probably won't work if you have 6 kids. Every adult who can work works in the Bay Area to get by so there is very little slack time in the system to barter with each other.

What little there is can usually be extended to 1 or maybe 2 kids. We routinely have 4 kids at our house (2 ours), but nobody will be able or want to manage more than that, and not very frequently. Even that sort of arrangement only happens via trusting friendships that are built over time. Distributing the load across multiple arrangements would be logistically daunting.

Perhaps a religious or cultural group would extend their resources in this way, but you would have to belong to those groups, and even there, nothing is free.

And one more thing: any place in the Bay area where you can find a house large enough to comfortably house 6 kids in a relatively safe area is going to be unbelievably expensive. I'd you live in such an area your neighbors will either have many fewer children or be wealthy enough that they hire full time childcare, so they won't have the same needs for village style arrangements as you.