Let's not get too nostalgic. As Duncan points out, Payton hit a similar lull and had similar locker room issues in '14, '15 and '16, going 7-9 each year for three years. When the Saints then opened the '17 season 0-2, the calls for Payton's head were loud and widespread.
I was told by a person socially friendly with Saints' execs that the issue was essentially one of veteran Payton loyalists undercutting Allen when things weren't going well. Not open rebellion, but a kind of locker room clique of bad mouthing and "back-in-the-daying." Names that got mentioned as being malcontents (at least from front office's view) were Thomas, Graham, Winston and at times Kamara, Jordan and Lattimore. This was seen as having a bad effect on younger players and setting a tone that the old-guard was kind of above the rules.
This problem is exactly why many organizations clean house when there is a leadership change.