In order to achieve bank levels of trust you need to be confident of being able to enforce contracts against the organisation you're dealing with.
This usually comes from them being vulnerable to an actor who you believe will wield force on your behalf if they break contract. This then means you're stuck trying to find an actor who
1. you trust to wield force on your behalf (i.e. they're not just some cutout established by your counterparty)
2. who you are sure has means to wield the force against the counterparty (perhaps they can prove that they know where the counterparty lives)
Those are difficult criteria to fill, which is among the reasons why over a long period of time, we've invented governments we can put up with.
I can imagine two other scenarios where you might be able to generate the level of trust you want:
1. If the counterparty can somehow prove that it's not in their short-term interest (because sometimes people run successful and profitable services for years and then abscond with the money suddenly) to just take all your money and run.
2. If you have insurance with an insurer that you are sure could pay you if everyone transacting with that counterparty made a claim at once. (But then, how can you trust that insurer unless they are vulnerable to someone you can trust to wield force against them on your behalf if they break contract?)
I think any future where bitcoin organisations are as trusted as banks is a future where they are registered with the government as financial bodies and regulated in such fashion.