I agree with the timing of the so-called "Black Monday" mass firings of NFL Head Coaches on the Monday after the regular season ends. If your team is going to take a new direction, there's no need to keep a lame duck coach around any longer than necessary. It would just be awkward and not productive.
And I'm OK with organizations immediately starting the interviewing process for replacements with coaches who are not otherwise engaged.
What I don't like is HC positions being snapped up during other teams' playoff runs. The Bengals are a perfect example. Both Gruden, the OC, and Zimmer, the DC, had multiple teams supposedly interested in interviewing them. All the while they're supposed to be preparing for their biggest game of the season. If Gruden was only at 95% focus and Zimmer at 99% (Zimmer is human, after all, and has aspirations), that little bit of lost focus or distraction could be the difference in the next game.
Practically speaking, being an NFL head coach is a lifetime dream for a lot of coordinators. How would they effectively block out thinking about it, at least a little, when the prospects are so close at hand? There are mixed incentives fr winning (but staying out of the job market) and losing (but getting to start interviewing for an HC job).
So what I'd like to suggest is a moratorium on HC hiring until after the Super Bowl. Then the coaches on the teams who have earned a post season berth, would be allowed to focus on one thing, the next game. Without the distractions. I realize that having to wait until February would set teams in transition back a month or so. Too bad. The lesser teams don't get to draft college players early to give themselves more time.
The complicating factor here is that coaches often move between college and NFL teams. I don't see the NFL asking colleges to wait until after the Super Bowl to contact coordinators or coaches of those not in the playoffs. So some plum college jobs would be long gone by the time the Super Bowl was over.