I am one that wanted an alpha coach because I watched it with Payton for so long. It's certainly not the only characteristic that a coach needs to lead nor is it even a necessity. As you say, the Saints have many alphas on this team in Thomas, Kamara, DeMario, Cam, Mathieu and even arguably Carr. It seems to me that Dennis Allen has the intelligence to put together a good defense but he doesn't exude a commanding presence that the guys I mentioned would run through a wall for.
I should say that the Saints need a leader at HC. A leader is needed for sure. Alpha or not.

I get it. The difference is that we see Alpha a little differently. I see Sean Payton as terse, introverted, and somewhat enigmatic; like a Tom Landry, or Bill Belichick. When I think Alpha it’s along the lines of Lawrence Taylor, Mike Tomlin, Dan Campbell, Lyle Alzedo, etc… and I don’t see Sean Payton fitting into that list very well.

Now if I contrast Sean to mealy mouth Dennis he is certainly closer to an Alpha personality.