I can do that with gold.

Gold is less volatile, and the value of the volume of a €2 coin in gold is 9 months of El Salvador nominal GDP per capita.

Or the gold could be stored elsewhere and accessed via an app, with exactly the sort of database every bank already uses.

Either way, BTC adds nothing. Apps might, depending on the banking infrastructure, but BTC itself? Nada.

Yeah, but you're still having to pass through Coinbase or any other exchange, and they can remove it from you there. That's the point, if Bitcoin needs an exchange to be functional you're losing all the differential features that Bitcoin has.

So the modern version of keeping gold under my mattress?

A bank by any other name is still a bank. Sure there is definitely a legal difference such as deposit insurance and access to their lender of last resort. For the customer however it doesn't matter they still need KYC, the cops can still see their transactions, they still have to ask where the money is from, they can still knock on your door and put your hands in handcuffs. Many of the advantages to people in El Salvador with BTC is avoiding extreme inflation in the national currency. However one does well to note that inflation woes in the Atlantic world actually correlated with a partial deflation of the Bitcoin bubble, so if you're in the EU/US bitcoin is an investment when money is cheap, not a hedge for your money becoming cheap.

If you need to keep your money on an exchange to be able to do transactions, you'll probably still need to go through a bank or exchange or the same thing by any other name.