Why do pools give rewards to miners who solve problems at a lower difficulty than the network?
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.