The cumbersome part of notarization is having to physically go to the bank, which would be the same for the post office. The performance of showing up in person, and showing a physical hard-to-forge ID, is exactly what drastically raises the bar for an attacker.
Doing this instantaneously implies skipping the heavyweight process. Which I assume means doing something like a one time (or periodic) cumbersome in-person process, and then repeating a quicker online process. Any private company could create such a system right now, bootstrapping off the notarization framework to do the heavy lifting.
But credit cards and other traditional financial institutions don't actually care about identity in such a strong sense. The standard process for setting up online access for a new account at a brick and mortar bank involves leaving the bank, going to their website from home, and entering your not-particularly-private information to sign up. That could be easily modified to setting up initial access credentials right in the bank if they wanted to, but the traditional financial system is pretty forgiving for the most part.
So we're really talking about custodians of new bearer instruments (eg cryptocurrency), which are more like cash. Hence my reference to these new holding companies being the ones that should be integrating notarization into their exceptional account access methods. Of course they could also just insist on the use of real security tokens, require a second one for backup purposes, and simply not use SMS at all.
FWIW another more convenient way of doing brick and mortar verification is to snail mail out a letter with a code on it to the address of record. It obviously doesn't have super security properties (anyone can steal mail), but it's much better than electronic-only non-verification and much nicer than surveillance database verification.