18-41 as a Head Coach

You act as if Payton took over a bad team. Not making excuses for Dennis Allen but he took over a team in salary cap hell and QBs of Andy Dalton, Winston And Taysom Hill. Featuring an all pro receiver who had basically missed 2 yrs after a ridiculous contract. Sean Payton was in a good cap position including having the #2 overall pick of the draft and not only future hall of fame QB but quite possibly top 5 of all time. With zero expectations following Katrina. One could argue Sean Payton had many luxuries Dennis Allen did not.
The Saints team you are describing that Allen took over in January 22 had just completed a winning season without Drew Brees and while starting multiple QBs, none of whom are current NFL starters. Why did they have a winning season? IMHO because of coaching and a strong roster except for the QB position.

Not many first-year head coaches inherit a team coming off a winning season, as Allen did.