I think reality is that the Saints weren't the most attractive job opening. Pederson and McDaniel, two of the names who get thrown around a lot on this board, both were hired by their respective teams before Allen was announced as Payton's replacement. We know the team was interested in Pederson and brought him in to interview. It's assumed, by some, that they weren't interested in McDaniel, but I'd need to see confirmation of that. I think it's at least a possiblity that an initial inquiry could have been made, but McDaniel had his own preferences to work through, and ended up taking the Miami job.
It's reported that the Saints also spent an extended time with Bienemy, and that they reached out to nine candidates and interviewed six of them.