Military, Political, and Enlightenment. There's no real need for Honor as a basic win condition if its tied to everything.
Acknowledging that in your model politics would replace both honor and dishonor, I'd honestly like to see a unified victory condition.
Like a new five rings enlightenment thing, where the different rings echo the old victory conditions, and it's first to 3 out of 5 wins.
2 of them Military oriented
2 honor oriented (or politics, whatever)
1 something else. Maybe reflective of enlightenment somehow.
That way pure mil decks can plan to win 2 and grab the enlightenment rings for the win.
Same for pure honor.
Balanced decks (perhaps known colloquially as "enlightened" decks) can grab any of them, but are not as good at Mil as pure military, and not as good at honor as pure honor.
I'm not sure if it would be best to have the rings be mutually exclusive in this model. i.e., once I get Fire, you can't get Fire, but I suspect it would be better if all 5 remain available to all players regardless. That way, the rules don't have to be different in multiplayer, and the Mil deck that gets its rings *first* doesn't auto-win.
I actually imagined the 3 out of 5 rings thing almost exactly this way. I think there's potential there. I really hope that FFG starts giving crumbs sooner rather than later.