I expect on-chain transactions will eventually be extremely high-value, between major players and institutions willing to pay the fees, while the daily transactions will be on sidechains/other layers.

I'm not sure the fees will actually change a whole lot if denominated in BTC. I just expect that when 1 BTC is worth a lot more than it is now in fiat, people won't be paying for their coffee on-chain (assuming inflation doesn't do crazy things to the price of a cup of coffee!).