I really don't see it as almost every single transaction will be on lightning. I see it as only the smaller amounts(probably 0.02 BTC and below?) going through lightning, whereas transactions above that will be made in the base chain.

Sure, probably less fees for the miners in theory, but if lightning does gain adoption, what would be the point of other payments-focused cryptocurrencies? They'd most likely head back to Bitcoin and use it instead, hence increasing the total daily transactions(both in LN and in the base-chain) somewhat balancing out in the end in terms off profit for the miners.

Though I haven't really checked out statistics of transaction values so yea.