This question already has an answer here:
I'm guessing that miners are randomly selected, but I'm curious as to which miner receives the fee for gas paid for each transaction?
In hindsight, the answer to this question is obvious: of course it is the miner that also solves the PoWhash. They get the block reward AND the gas.
The suggested duplicate answer (the questions are also very different actually) does not hone in on the PoW incentive and why a miner is rewarded. The answer for this question targets exactly how the gas is rewarded to the miner.