If anyone is interested in learning more about this whole transaction dilemma, you can read this
topic. To make a long story short, there are no shortcuts in constructing a block. Transaction-biased blocks earn less in the long run. The idea of letting miners create their own blocks is also absurd and unfair because they could be including their own low-paying transactions or providing an acceleration service and thus earning more money outside the ecosystem while hurting their co-miners. The fairest and most profitable approach would be to use the default GBT and not provide any acceleration or if you insist on providing that, you should compensate the miners for the lost opportunity.
Another wrinkle is that there may be payment for inclusion outside of the transaction fee, and it may not be obvious to the miner who delegated hashpower to the builder (via GBT, for instance, by default). See e.g.
https://arxiv.org/abs/2110.11740 (“Selfish & Opaque Transaction Ordering in the Bitcoin Blockchain”) for evidence of pools acting in this way.
So using GBT (at least by default) may not be enough to ensure fairness of transaction selection, even though this is orthogonal to the question of who builds the block. (And I still think it leaves open your cominer fairness point in a shared reward pool.)
