If hodl invoices achieve wide adoption it will have an outsized impact on the network. A lot of things will change but let's keep it simple: it will push the entire network toward a reputation based system. This is bad for a permissionless network. Ideally, I should be able to spin up a new node, commit capital, and send payments with a high success rate. But now everyone is suspicious of unknown nodes. They're tracking your invoices to see how long they stay open. People run block lists and maintain a database of surveillance data. We already see this happening with channel probing. Maybe it is inevitable, idk. But I certainly don't agree that we should run toward this outcome. ¯\_(ツ)_/¯
In my previous reply I was facetiously describing ecash, which IMO is a categorical improvement for the async payments model. The sender does not need to be online for the recipient to receive payment. This is the role of the central server or "mint". It's just wheeling and dealing lightning payment promises all day, blind to the identity of its users. With the advent of DLEQ proofs, you can even trustlessly transfer ecash offline as long as the recipient knows the public key of the mint. People complain that the mint can rug you but is this really different than when a hodl invoice expires and you get a channel force close? I contend that it is not. This design adds a central party to facility async payments, improves privacy, and doesn't push negative externalities onto the network at large. The only downside is that you can't yet hack something together in a weekend. Or maybe you can...the gap is closing fast!