We could reach a point where the vast majority of bitcoin transactions take place on Lightning, with people keeping their channels open pretty much permanently.
I don't even think that would be such a bad thing. I now trust my "local" bank with much more money than I'll ever put in a LN-channel for small expenses. If my bank would offer a custodial LN-wallet with the same level of trust
*, I would use my bank for LN-payments just like I now use that bank to pay bills.
Of course, that gives the risk of fractional banking on LN, but that's no different than what crypto exchanges can do already.
Or, as ETF says, it could be that nodes and miners start accepting transactions with fees of less than 1 sat/byte, you just round the fee you are going to pay to the nearest whole satoshi.
That won't work for mass-adoption without larger blocks.
What would happen to fees if the minimum amount on-chain was 1 milli-satoshi/byte ?
Fees would probably get lower than 1 sat/byte once in a while, when blocks aren't full.
My use cases for higher decimal precision are required
today 
My team are working to get IoT devices performing instant Lighting payments in an M2M network of fire sensors, alarm sensors, humidity etc etc.. These devices pay each other as and when they want a reading from any of the other senses. Thousands of sensors thousands of times a day, thousands of locations.
We already need lower than satoshi precision, and the idea of
rounding up.. lol.. this is Bitcoin.
LN allows payments as low as 1 milli-satoshi already, so you can do this already. I assume you won't settle channels for thousands of transactions on-chain too often, right? And when you do, the fee will probably be far more than 1000 satoshi, so why would you care about a fraction of a satoshi rounding difference at that point?
* I don't trust banks at all but I have no choice.