So in contrast to a fiat bank, where you keep your funds permanently, you could use a "lightning bank" just to hold last nights payments and transfer them to yourself every morning. Or you automate the withdrawl to take place every hour.
And: Couldn't your phone be your server? The way I understand LN, receiving funds just means to exchange a few bytes of data. Can't phone apps receive push messages?
if we hadn't had NAT and reverse tunneling then things might have evolved differently. maybe a way to push data to phones over UDP or low power servers that could wake up to handle single requests. but it didn't play out like that.
Receiver of funds are normally merchants, and merchants normally already have payment systems that require some kind of connection anyway.
like Chivo? take out the middleman(Banks) and put governments back in control of the funds.
bingo :)
Think about it...
Imagine if you could have complete custodial access, control and oversight of the details of your citizens financial info? No bank accounts, no stash of gold. It all through the US-backed custodial wallet "Trump's Coin purse". Everyone download it now and get $30 free! Also, it's a requirement for all employeers now so, deal with it banks.