In fact, the "fighter" detail is the deciding factor. You can persuade the Landsmeet, but if you can't beat Loghain in the duel, you lose. The Landsmeet follows your recommendation on Alistair because you or Alistair have proven yourselves capable in single combat.

If that's the case, then shouldn't the PC Warden, Wynne, Zev, Morrigan or the mabari be the leader of the country if they win?

I'd totally be rooting for the dog. :)