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?

So at minimum the proposer fee would cancel out all the rewards received via attestations, and by proposing honestly, validators earn back some or all of that fee, plus whatever they receive in EL tips.

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