Unfortunately Bitcoin ASICs are only good at double SHA-256 hashing; meaning it only does sha256(sha256(INPUT)). Probably not remotely useful of any modern password systems.

Correct, the boards are not useful for anything other than bitcoin mining. However, you give an engineer that form factor and have them design a password cracking board, now you pull out and scrap all the Miner boards, replace them with Password cracking boards, and all of the investment in the infrastructure around the boards is preserved.

My reasoning is that replacing just the boards should be less expensive than creating the entire system from scratch. It may not be much less if the infrastructure is cheap compared to the boards but less.

So... any major government site is a target? People use sha256 because it's easy and simple.

Bcrypt? There be dragons!