I don't think that his tenure rests on finding a QB that's just asking too much heck even Payton failed multiple times as far as that goes so it would unrealistic to expect Allen to just spend his time solely searching for a QB.
He needs to win and I think to do that you need to fix the wr and tackle spots and taking a risk on taking a QB just sets you back at least 2 years. If Allen tanks for 2 years because he didn't fix positions that need immediate solutions he will be ran out of town for passing up the chance to get two players that could immediately help and instead drafted a QB that is sitting on the sidelines while we are scoring 10 points a game.