The franchise's big issue over the last few years is stability vs incumbency. While our FO has been stable, it has also - even with some degree of minor turnover - been remarkably incumbent; our stability is from utilizing known coaching/FO assets that are familiar with the "culture". The bonus of this is that your franchise has known and valid quantities in charge. The price for this is a lack of opposing voices, which is best (worst?) manifested by an inability to course correct or properly self-examine one's deficiencies. The perfect example for this is Dennis Allen's rhetorical "keep chopping wood".
The NFL is small world, in the end. Coaches and staff are recycled endlessly through the teams and the league (as a whole) is remarkably conservative. While you don't want your team becoming the Carolina Panthers and jettisoning staff and core vision year after year, you also cannot afford to value stability over development, or you turn into the Saints and Patriots. I think Sean Payton understood the structural limits of the Saints and recognized he had hit his limits here and that the team was becoming an echo chamber (which is, to a large extent, a natural evolution after long-term success). Some corrections were made by bringing in new faces for talent evaluating but that has not been enough. We are now stably mediocre. We won't get better until the front office recognizes these limits. And we are not there yet.