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.

028559d218

ya, i like this. i mean if a user already is perhaps paying for a channel with LSP, seems like an add on service could be to bank payments for users until they come online as part of their liquidity buy.

BeeAye

I can’t find it right now but this is planned as part of the LSP/async payment spec afaik

k00b

bitcoin

Falkvinge Criticizes Lightning (2018)

standardcrypto

My solution would be to have payments go to bank-custody wallet with high uptime, and drain funds immediately to self custody whenever possible, ideally it could be done within seconds. 