Why wouldn't fees go all the way to zero? Zero times a zillion transactions is still zero.
Ofc there will be miners who make much larger blocks no matter what, but it comes down to how urgent is your tx?
There would be an equilibrium where the fee is worth the marginal cost of mining it
shawabawa3 already explained it "There would be an equilibrium where the fee is worth the marginal cost of mining it"
The miner has cost X and wants profit Y. He adds those together and now the price is X+Y. Someone wants his transaction on the blockchain, he is going to pay X+Y unless there is a miner that offers an even lower rate.
You're making the assumption that the block reward is high enough to make transaction fees irrelevant, which is a trivial observation and not what we are interested in talking about here.