Yeah we live in interesting times in which the historical role of banks and what banks actually are used for are diverging.
There's two kinds of things here (and I'd lump them separately):
(1) there's "putting money in vaults" and handling long-distance transactions (eg: debit cards, check-clearing, and so on)
(2) there's banks as "so you've got money you want to lend but no time? give your money to us and we'll loan it out, passing some share of the profits back to you" intermediaries
(1) is arguably what most people actually use banks for; (2) is how most consumer banks make their money (in the form of mortgages and cds and so on).
Aside from historical circumstance there's no particular reason why (1) and (2) are handled by the same institutions. Since (1) on its own isn't profitable any institution that does (1) needs to have its (1) activity subsidized; the present situation is that banks subsidize (1) via (2), but there's nothing that intrinsically says institutions that do (2) should do (1) (or that the only way to subsidize (1) is via (2)).
In many countries (Japan, some european countries that escape me now) there's an entire system of "postal banking" which is basically de-facto state-subsidized banking expressly for the purpose of (1).
If you think about it the connection between (1) and the postal system is pretty logical: the postal service exists to streamline communication within the state; it's pretty sensible to piggyback a payments and money transfer system on the existing communications infrastructure (and of course it's much more reliable to transfer money then it is mail an envelope with cash).
The USA never went the route of establishing a postal savings bank (nb: the historical names include "savings" but they always allowed payments / de-facto "checks" to be drawn); this was mainly b/c the idea was invented well after the founding of the republic and the USA is pretty slow on the uptake wrt good ideas invented elsewhere.
One of the takeaways or realizations I see people coming to a lot in response to the current crises is along the following lines:
- at the moment keeping commerce flowing depends on keeping checking and debit and credit-card systems all working smoothly (in other words: keeping the (1) activity going); without that the economy halts
- currently keeping (1) going is dependent upon the health of various private parties (banks + other financial firms); as these entities don't really make any money from (1) directly their ability to perform the services of (1) is contingent on how well they're doing in their other endeavors (activities in category (2), writ large); (1) is not enough
- thus disruption in (2) leads to disruption in (1) leads to disruption in the entire economy, even when (so it seems) there's no direct connection between (2) and (1)
...which leads to a realization that the current arrangement worked until it didn't but it's not that great of a system because it needlessly couples (1) to (2); depending on the kindness (and prosperity) of strangers to keep essential infrastructure running isn't a good plan.
Once people have that reaction they tend to produce one of the following proposals:
(A) set up a system of state-run full-reserve checking+savings accounts + debit cards, etc. (and optionally: phase out the FDIC guarantees; if you want a guarantee go with the full reserve bank); fund operations from taxes
(B) set up a dumb/simple/basic bank that just does the basics and takes almost no risk
I tend to think (A) is the longer-term smarter option but am well aware how against the grain it runs in the USA; (B) is like someone trying to accomplish the same goals but not really understanding how the tools they have at hand work (that's what the original post is like).
The quick case for (A) is that there's a reason the constitution grants the authority to build a post-office: aside from the general betterment of the commonwealth it specifically prevents a situation where your (then-fledgling) government is effectively held hostage by private-sector actors because those private actors own and operate infrastructure essential for the government to function (what good is a federal government if Megamail Inc. won't deliver it's mail?). These days basic savings + checking + long-distance payments are critical infrastructure on par with the post office (and on par with the original grant of authority to coin money and so on), and to avoid being held hostage the right option is to provide a de minimis payment infrastructure.
Good luck with that politically, though.