Does a logical mining pool assign harder and harder difficulty to the same miner?
[Edit: my previous answer was incorrect in it's explanation, I've updated it since]


  would the next logical thing for the pool to be to assign that miner a slightly harder bit of work to do?


No. In a simple sense, consider this: if the difficulty is increased every time a miner submits their share, then it will increase endlessly over time, which is obviously undesirable. 


  If the pool does not assign harder work and lets the miner continue to solve problems at the same difficulty (which is less than the network difficulty), how does the overall pool benefit?


Miners in the pool will split the rewards for valid blocks amongst themselves, according to the share of hashpower each miner is contributing. The pool difficulty is a way to track each miner's share, while also dividing up the work load amongst pool participants. 

Each miner has a certain amount of hashpower available, and will set their difficulty to an appropriate level so that they can submit their work and claim their share of the rewards. Setting the difficulty too low is not a way for a miner to claim an unfair share of the rewards, it just means they'll likely find a valid hash sooner, but that valid hash will be worth a smaller share, which balances things out.