> Couldn't they just get these benefits (predictable fees, fast settlement) by ... running a database between these financial institutions?

Sure, but they wouldn't get all the legal and regulatory bypass benefits of using cryptocurrency.

Is this like Uber and AirBnB all over? They don't want to be a bank just like Uber doesn't want to be Taxi (means following regulations about insurance, accessibility etc) and AirBnB doesn't want to be a hotel nor a landlord?

In some countries, regulators simply point to these companies and say "Ok, so you're driving people around for money thus you are a taxi? How would we not regulate you as a taxi?".

And this should apply for a bank or financial institution that tries to avoid banking regulations through technical means, no?