Lightning isn't an IOU, its actual Bitcoin. What you are doing is trading payment states in channels, these payment channels contain real Bitcoin, locked in smart contracts. You can settle (and close) for your channel for Bitcoin any time you like, and there is a punishment mechanism if you try to cheat (steal) and replay an old channel state.
This style of network, or a higher order version of it will certainly be the way that we further decentralize our economy and scale to every human and system. It is nonsensical to store full transactions between everything in an immutable ever growing data structure for all time. You can condense what needs to be stored forever in channels, or go further smaller still and store commitments+proofs.
We need to move past the blockchain to store all the things model. It was cool when it was small, but we need to be thinking about how to implement higher order payment or economic concepts rather than raw txs.
Finally, to your point about businesses not trading in third-party checks... Kinda! When the store takes your payment over the PCI network, the card holders bank and the vendors bank trade some payment state between each other, and delivery of funds can take days to clear. Its all businesses trading slates, e-checks, e-balance sheets. In essence lightning isn't too dissimilar to how things work now with electronic $s.