Yes. Their reputation is taking on the reputation of projects they support. If they want to keep good reputation, they should watch over their sponsored projects.
And for what its worth (very little), I have worked at this point at various YC startups over close to a decade now, and it can't help but be a quite negative signal to me as non-CEO, lowly dev worker. I get that they have a bunch of successes under their belt, and this is near superstition on my part, but the heuristic of staying away from YC companies has made me a much happier employee.
Also, might your argument require that YC's leaders eventually gain access to a time machine, so they could advise their earlier selves against supporting projects which later proved disreputable?