So, you expect their society to take the word of a villain participating in what is likely the world's biggest villain attack in the history of what we know within the series?
Because this is just Dabi's flavor of hypocrisy that Toga exhibited in that they're all just a sad, misunderstood lot that are only doing the things they do because the rest of the world made them. Dabi had a bad father? Why yes, let's murder people because of it! Heroing != Parenting, and while Endeavor may have striven to be #1, it also wasn't solely for fame.
That said, Endeavor also didn't inherit the #1 spot under the best of circumstances. To say it really matters in the grand scheme is also rather pointless since big heroes are more about their agencies and the related work they do. Nothing about his performance against the Nomu or Shigaraki would also imply he is unworthy to continue heroing. He'd probably even pass Stain's twisted litmus test.
Now, could he die? I suppose so. We also thought All Might was going to die against AFO. In comparison, however, I think him living would be better for the narrative since the groundwork for redemption has been laid out for a while now.