They do look at bank accounts, but it wouldn’t be hard to fool the system because they’re counting on you to be honest and show them all accounts, not just one that is handling pin money.
Yes, however, all bank accounts are reported to the IRS and, by extension, using IL as an example, to the Illinois Dept. Of Revenue. I’d think that the IL Dept. of Human Services can check with IDOR about, at least, previous years’ interest income, which would then give them a pretty good idea about one’s account(s) levels or other such resources.
Granted that I do not know how often IL-DHS or similar agencies in other states actually do check. But on the other hand, I was brought up to be honest...