Yup. The actors don't even need to trust each other that much, because they each still need to use their public key to execute a transation. The level of trust is just that everyone will play nicely, such as not purposefully ignoring transactions.

The Bitcoin model is good at incentivizing actors to play nicely. Banks have no need to create this incentive, it's already in their interest. The Bitcoin mining system also solves another problem: how to evenly distribute new money in a fair way. Banks have no need for that either.