I think people will use a combination of self-custody and on-chain. On-chain for savings, like a savings account.

Lightning for checking/daily spending... and when the channel is depleted people will refill it from an exchange or bank-like entity. The same thing which is technically possible today. Ie a lightning withdrawal from the exchange/bank to a self-custodied lightning wallet. I also believe that liquid will play a role in this (with atomic swaps) as a safe(r) intermediary between lightning and the exchange/bank if necessary.