I think what OP is saying is that onboarding all of humanity onto the lightning network would exceed the capacity of the Bitcoin network.

Yes, once everyone is onboarded onto lightning the 1,000,000 tps bandwidth would apply. But to benefit from that, at least one payment channel per person would have to be written to the Bitcoin blockchain—a process that would take at minimum 30 years if all of humanity is to be on-boarded—and would more likely take several centuries, considering that those payment channels would also be competing with other bitcoin traffic.

Also consider that the security model of the lightning network requires that fraud penalties need to be written to the blockchain within a certain time window if a fraud is to be reversed. A clogged Bitcoin network where transactions are waiting too long to be added to the blockchain—or where blockchain fees are too high to justify adding a fraud-penalty transaction to the blockchain—would result in the entire security model breaking down.

Most small transactions would be via lightning channels between exchange providers. Basically, whomever was running the backend of the wallet app would be an exchange provider. In an ideal world, people would keep their long term savings in other instruments like bonds or even gold and keep day to day cash in exchange providers. Yes, they would be trusted, but not a lot. I think the fees on the Blockchain would be substantial enough that most people wouldn't be using it for coffee and it would only be for bank to bank transactions of these exchange providers.

7 transactions a second would mean 604k transactions a day which is a reasonable number for major counterparties with more than about 35 btc each. (21M/604k)

The craziest part is that no other cryptocurrency has this problem. It is entirely self inflicted that bitcoin has 1 KB/s throughput for the entire world and everyone else just didn't cripple themselves.

Does everyone need a payment channel written directly onto the btc blockchain?

There’s no theoretical limit to LN transaction numbers because there’s no shared global state. You could have many very high throughput channels processing payments in parallel.

Any transaction still has to make it on to the chain. A side chain would be like us playing catch in the back yard and thinking we will somehow be padding our major league baseball stats. It doesn't matter until something happens where it counts.

If you find someone to play catch with, that's great, but the whole point of a global transaction network is that anyone can send a balance to anyone else.