As long as BTC transfers between custodians are possible, that should be quite small problem. There might be winner in terms of liquidity and volume, but because anyone can quite easily fire up custodian service, there should be also plenty of competition.

So we have basically invented... banks, but with bitcoins?

With Bitcoin, I don't have to keep my savings in the bank even if I choose to use a bank for other services. As someone whose had their bank account frozen before, this is of great value to me.

I could use banks or Lightning like a checking account, while keeping the majority of my savings under my control.

Yeah but you are most probably using a centralized exchange to execute BC transactions for free which might froze your account as well in the future because $REASON. Also thinking that in the long-run, in a future where BC is accepted almost universally this is not going to be heavily regulated by governments it's being naïve IMO.

Yes. Financial institutions are useful for fiat and non-fiat currencies, in fact they were born long before fiat currency.

The full circle