Part of taking care of things is applying for any benefits due the surviving spouse and getting funds into their name alone. In my experiences helping settle affairs with family, these things can take place quickly.
Financial institutions can be slow to move to recover debt, if they even try at all. A family member took out a second mortgage with the bank that had his checking and savings accounts. He died, still owing a large balance on the mortgage. The bank had every right to seize all the funds in his accounts. The money wound up going to the state as unclaimed funds where they were subsequently recovered. This didn’t make sense, but it happened.
Agree that they will find out, but sometimes they don’t take the expected action. 