How does Lightning actually solve high-volume transactions when everything eventually needs to go back on the chain?

I’ve been rolling this around in my head and I don’t get it. One of Bitcoin’s weaknesses is long confirmation time and limited block space, thus limiting throughout. Lightning is meant to solve this so settlements happen on the L2 but all this needs to eventually be settled back on the L1. I’ve heard it explained that if, for example, a supermarket transacts in bitcoin it won’t settle, again, for example, until the end of the day meaning only one sum transaction is needed. Don’t they still have to settle with millions of people? Are they settling with banks and other institutions?

If, instead, we live in a future where the monetary system is built on bitcoin then we may transact in some other digital currency and true final settlement occurs between institutions and banks. Don’t those banks potentially need to settle with millions of people also, and if not aren’t they the custodians of Bitcoin owned by others, aka a similar centralised system that Bitcoiners fear where you also don’t “own” your coins. Is this the future bitcoiners are happy with given its technical limitations; billions of people, not so many millions of transactions on L1 per day. Have I missed something on the technical side, or maybe my conclusion is wrong? If one billion people us Lightning, in theory they may want their Lightning wallet Bitcoin confirmed on the L1. How are they every going to get that without insane fees or time?