Fixed that for you. The big banks have one thing going for them, they're big. That means they will have a wide network and a large service organization, but most feel no need to meaningfully compete for your business. You're going to use them because they're big. You're not going to get meaningful interest on your deposits at a big bank, because they don't need your deposits; bankrate shows me rates up to 4.75%, with banks I've heard of at 4.3%, Ally at 3.75%, my credit union at 2.5% (at least it's moving up, although the pace of increases is a lot slower than the decreases were :/), and Chase is at the national average of big banks: 0.02%. Certainly, 0.02% was understandable in the zero-rate environment, but I'm pretty sure it's been Chase's interest rate for the last 30 years (no data, just a hunch). Mortgage rates float in and out of competitiveness, especially if you jump through the hoops for a relationship discount; that's a good business where volume means profit at origination, and it's easy to sell the loans if they want to (that's part of why their underwriting is so cookie cutter; gotta make it easy to resell).

If you're in the SF Bay Area, you should really check out the local credit unions; some of them are pretty decent. Co-op/shared branch banking takes care of most of the access issues, but you might want to be aware that after hours telephone service is directed to a shared branch call center. My credit union holds and services the mortgages they originate, which means you don't have to deal with the servicing changed dance, and also they're able to do limited paperwork rate adjustments for a reasonable fee; much simpler than a refinance.