Not a finance guy but I think that would literally be the opposite of how banks work. They have to lend out customers money in various forms, including giving that money to other customers in the form of loans, loaning it in huge chunks to companies for larger percentage rates than they give customers who keep their money in said bank, and so forth. Not sure how a bank that kept every customers money on hand all the time would work.
Well, there are banks that charge you money to hold onto your cash for you.
A "narrow bank" could work, but the Fed currently denies them banking licenses, since (presumably) the social function of banks lending out deposits is considered useful.
The Narrow Bank proposed a structure like this, where they would take deposits from customers and just park them at the fed, passing along the fed rate minus a small cut. This is more or less perfectly safe: the Fed basically defines what a US dollar is, and cannot be insolvent.
But the Fed won't let TNB open an account. The basic reason seems to be that they worry that this model is destructive to the US economy, which relies on banks making loans so people can buy houses and cars, and businesses can operate. If everyone banked at a narrow bank, the economy would seize up from lack of capital.
Whether or not you agree with this take, the reality today is that you cannot run a bank like you've described in the US.
(This is all covered in more detail by Matt Levine: https://www.bloomberg.com/opinion/articles/2019-03-08/the-fe...)
Or just don't mess with money that belongs to customers. Be the world's first reliable bank.