Living in Poets' Corner
My wife and I moved from NYC (UWS) a year ago last June due to a similar work situation/opportunity. We are in Poet's Corner in Berkeley and it is filled with families and young ones. Strawberry Creek Park and Hidden Cafe are neighborhood favs. North Westbrae is also a great area as well as neighboring Albany.
There have been a lot of SF transplants into the East Bay and Alameda County due to the pandemic (three moved onto our street alone in the last nine months). Although we rent, we did attempt to dip our toe into buying but homes are being overbid $400k-$500k with all cash buyers from Silicon Valley. SF is the only place where sales and rentals have dramatically dropped (similar to Manhattan) and you can find good deals. But depending on where the work is located, do keep in mind traffic as it tends to be frustratingly bad. Public transportation is doable, but again due to the pandemic has gotten even worse than usual. If you do move to Berkeley or anywhere in the East Bay, a car is needed.
Rents in Berkeley did drop since last March, but now that UC Berkeley is going back into session come fall, rents are going back up. Also due to the school cycle, much doesn't come to market until May/June/July. We tried looking for rentals in March/April and found next to nothing. Luckily we were able to stay a bit longer in company housing until we landed on something.