Why do pools give rewards to miners who solve problems at a lower difficulty than the network? I'm trying to understand why a pool would want to give rewards to miners who mine at low difficulty.  Please tell me where this logic falls apart.  If the network difficulty is a nonce ending in 100 zeroes, but a miner solves a hash (not sure what words I'm supposed to use here) with a nonce ending in only 10 zeroes and submits that to the pool, the pool now knows that this particular nonce ending in 10 zeroes is not valid so other miners don't need to work on it, and as such the miner who just solved that problem has saved others some time, so they are entitled to a reward?

Ultimately what gains the pool any rewards at all is if someone solves the hash matching (or greater than) the network difficulty so I'm trying to understand how solving a problem at a lower difficulty helps the pool (and thus leads to rewards for that miner).
It's not that solving a problem at a lower difficulty helps the pool, it's that trying to solve the problem helps the pool cause you can 'accidentally' mine a block on the actual difficulty. Pool members share the lower difficulty blocks to prove they tried.

A pool member can find a block on the actual difficulty and don't share it with the pool on purpose, but won't get any reward by doing that.