Remember that the concept of the Inquisitor does not exist in Fade Rift. The Herald died at Haven before she was proclaimed Inquisitor and accepted authority to lead the Inquisition.

Canonically, in game, there has never been a single leader of the Inquisition, and if there was a single leader, the most logical choice would probably be Cassandra, its founder. (Multiple characters, including the Inquisitor, ask her in the game why she doesn't take command, and the answer is basically that she doesn't want to).

Just because the advisors are in a stalemate at the moment doesn't mean that they will remain that way, or will be unable to find a compromise. In game it's worked well for them up until this point - this is the first time they've had a real disagreement on anything, and there's no reason to think that they'll be unable to work it out, or that a council will necessarily do any better or incite less conflict in the ranks.