re: The argument for keeping Dennis Allen

There is no valid argument for keeping him anymore.

2 straight seasons missing the playoffs. Yes, he technically improved this year, finishing with a winning record; that's negated by losing every pivotal game they played this year. All season long, the Saints were in position to seize control of the NFC South, and each time they stumbled.

He's supposed to be a defensive guru (so they say), but defensive collapses cost this team several times.

Most importantly, he's completely lost the team's respect. At a point where the team still has hopes of making the playoffs (having done their part in winning today), the team gets a late interception and the ball on the 1 yd line. He states in the press conference that the players asked if they could get a score, and he said NO.
Team lines up in victory formation... and runs for a TD anyway. The backup QB ignores his call, and calls his own play. The first year RB who has not produced all year runs it in. The senior defensive end makes social media remarks that they should have gone for 2.
And instead of handling this inhouse, Allen airs it in the postgame conference.

That is simply NOT a leader, and no way you can salvage things going forward.