Cash on your checking accounts (or other cash-based accounts) can actually be difficult to retrieve when a bank goes under. The reason is that the money in your bank account isn't actually your money. It is a loan you gave to your bank. And recovering loans from a bankrupt entity is difficult by definition. Being unable to pay back all the loans is the very definition of being bankrupt.
This is further complicated by the fact that no bank in the world is ever physically able to pay out all the money in all the checking accounts they manage. The reason is fractional reserve banking. A bank is only obligated to keep enough cash reserves to pay out a small fraction of the money in all the checking accounts. Which means even the most well-managed bank never has enough money on hand to allow every customer to close their accounts at once.
However, in practice there are laws and regulations which protect private customers of banks and ensure that at least they get their money... eventually. The interim phase of a bank going bust might indeed cause some disruptions in the regular processes.
Financial assets like ETFs, on the other hand, should not be difficult to transfer to a different financial institution. They are not owned by the bank. They are owned by you. The bank is just holding on to them for you. And fractional reserve banking does not apply to financial assets like funds. When the bank claims that they are holding 100 index fund certificates for you, then they must actually have those 100 index fund certificates. Not just a fraction of them like it's the case with cash.