This is an unsolvable problem. As good a job as you're doing, YC is the sponsor of HN and just because of that there will always be this perception. The only way to get rid of this issue would be to completely divorce YC and HN, something that may for a variety of reasons not be possible and for a whole pile of other reasons not be desirable.

You're doing an absolutely super job, probably far better than most or all of us here will ever realize simply because moderation when done properly is all but invisible so don't sweat it, this is a thing that is as far as I can see not solvable in the current set-up. Those lines were drawn long before you showed up and within those lines you're doing the best you can.

FWIW I too recall several instances of users that were banned imho unjustly as well as some threads where the pro-YC bias broke through but over the vast amount of content generated here those are very very few instances, not by far enough to claim systematic bias or to be used as evidence for some nefarious plot. More like genuine mistakes and things done in the heat of the moment. And on later reflection some of those were reverted.

(If you want I can probably dig them up for you but you're busy enough as it is.)