Yeah, I feel like this Wikipedia graphic puts it quite well why that was rejected:

You see that “Pleistocene” vertical bar? And you see that tiny sliver of “Holocene” at the top. Yeah, the Anthropocene folks were basically arguing that so many riveting things happened in the Holocene already, that we need to declare a new epoch for what’s happening now.

Besides, if we do continue to irrevocably fuck Earth and the current mass extinction event continues to wipe out a big chunk of life on Earth, then a future sentient species might declare our entire existence as just the geological event that ended the current era (Cenozoic).