LN is not a magic solution, it only relieves fee costs and decreases onchain transactions if used wisely.
Which in LN case is multiple transactions between the same parties over and over again.
Once LN becomes mainstream, I can imagine the people who store their funds on exchanges, won't have any problem using the same exchange to make their payments. Currently thats not a viable option because withdrawal fees are much higher compared to using your own wallet.
I imagine a future in which exchanges keep a couple million dollar worth of Bitcoin in their LN hot wallet, and many people use the exchange to make their small payments without ever needing to fund a LN channel on their own.
Say: paying from exchange to VPS provider, paying from exchange to buy a coffee, or paying from exchange to fund a balance on website x. The exchange, VPS provider, and website x will run their own LN-node. The coffee bar will most likely use a large payment processor that runs their own LN-node.
There are already custodial LN wallets, but they still need to earn their trust. If a large exchange would provide their own convenient (mobile) wallet, I'd love to use it!
Is it really Bitcoin in which you and only you are in full control of your private keys? No.
Do I care for small amounts as long as I can make small payments without paying several dollars in fees? Hell no!