Validators derive a significant proportion of their yield from proposing blocks - both proposer rewards on CL and tips on the EL. Since this leads to increased reward variability (which hurts small stakers) wouldn't it be better if there were no CL reward for proposing, or even a fee?

This is indeed a way to reduce variability (I had a section on it for my variability post that I skipped due to space constraints). There are also some nuances to consider:

This is indeed a way to reduce variability (I had a section on it for my variability post that I skipped due to space constraints). There are also some nuances to consider: https://ethresear.ch/t/properties-of-issuance-level-consensus-incentives-and-variability-across-potential-reward-curves/18448/11