I don't think Anthy would have stayed. Even if Utena was dead, she'd come out of her coffin and I don't think that she, as a whole individual, would be able to stomach Akio's games. She left to find Utena, but even if there is no Utena, Ohtori doesn't seem to be the type of place you go back to if there's nothing there to call you back the way there was with Ruka and Mikage/Nemuro.

It is a romantic notion, the searching for a loved one after they've gone forever just because you don't know for sure that they're dead.

Whether Utena is dead or not is a moot point.  The point of the ending is to show that Utena really did achieve Revolution and the most direct example of that is Anthy choosing to leave.  The entire epilogue shows the effects of her achieving Revolution.  The Revolution was simply change. 

The conversation between Akio and Anthy when Anthy decides to leave proves that.  Akio is speaking about how he was going to start the duels up again, the implication that he'd been running the duels over and over again for years.  Anthy stops all of that by choosing to not play his game anymore.  Without Utena, had the duels gone on just like all the ones in the past, Anthy would have gone back to the playing the part of the Rose Bride just as she had before.  But because Utena changed things, achieved Revolution, so much had changed in Anthy as well as the other duelists be they seitokai or black rose.

I agree with (I think everyone?) that Anthy would have left regardless.

If she had chosen to stay in Ohtori for whatever reason it wouldn't have been as the Rose Bride and I doubt she'd have anything to do with Akio anymore. She'd still feel sorry for him, but he wouldn't be able to bring her around to his side again.