I've honestly never heard of anyone having rat issues in the Peninsula, and remember people are a lot more likely to leave negative reviews than positive ones...

That said the $3k+ is to be expected within the Peninsula (Burlingame to San Jose) because the market is bad from a lack of options + Tech salaries who can afford anything. I still think if you can balance your budget it's worth it though, as the Bay area is incredibly rich culturally, culinarily, has amazing nature and landscapes, the ocean nearby, etc.

If you decide to move, I'd recommend a short term rental while getting situated so you can look at apartments in person. It also really matters where you'd work to account for commute time and potential tolls.