Welcome! As you've quickly realized from other replies, it's very expensive to live hear. I'm up near Oakland and even that is expensive.
Commutes pretty much suck at all hours. There is simply too many people here. If you can, I would consider a smaller place closer to work - though by the airport is kind of a sketchy place to live.
On my comment about being close, think about how much time you could have to be with the family vs not when considering 60-90 minute commutes each way. Depending on where you go, that might be reality for something as close as 20 miles away. Keep that in mind - if you move too far, the house might be great but you could also end up not being able to watch the kids up and all that.
One thing I would add given that the OP is triathlete, is just pick where your family needs to live and just ride to work. The shoulders in San Jose are awesome (my relative perspective is riding around in Taipei at rush hour during a biz trip with my folding bike), and you can ride all year in shorts. No reason really to even get in a car, unless you have to use your car for work to visit customers/partners/suppliers/investors etc.
So if you choose to ride, you can take the commute out of the picture.....60 min ride each way is 18+18 miles per day.....10 hours of riding per week, 300K per week, zero time in the car, pick a slightly cheaper place to live, spend zero on gas, get extra training in. Everyone wins. If I lived in the Bay area this what I would do. It baffles me how few people ride to work there compared to my Greenland Iglooland deep freeze zone in Ottawa Canada where tons of people bike commute in crap weather. Almost no reason for a triathlete with teenage age kids to not be able to leverage the bike to get to work. It's not like you need to drop kids off and pick them up from daycare.