These are called narrow banks or full reserve banks. No such banks exist. There are a few reasons why:
1. It's not appealing to the market when fractional reserve banks get bailed out, as just happened with SVB. Why would you turn down interest payments when they're risk free? For narrow banks to make sense governments have to never bail out depositors.
2. It's not appealing to the market because depositors would be forced to pay into the FDIC scheme, but would never need to call on it, so they'd be subsidizing other people's risk taking and interest payments.
3. It's not appealing to the market because your balance would go down instead of up, and see (1) and (2)
You're right that full reserve is the only system that makes sense. People would still lend and invest, they'd just do it through funds that expose liquidity constraints by saying you can't withdraw from the fund without N days of notice. Banks already offer such funds, it's nothing new.
Unfortunately the system is stuck in a catch 22. To fix banking you first need to convince people you'll let people suffer when a bank collapses. The moment a bailout happens the whole concept collapses, and every full reserve bank with it. Yet such bailouts are controlled by civil servants who fold the moment someone starts to cry, they have nowhere near enough fortitude or democratic mandate to do this. It'd require a party to campaign on fundamentally reforming the way central banks work. In theory the Republicans could do this in the USA, but in Europe the ECB is controlled by the EU and outside the world of democratic politics.
