How many candidates would have to turn them down before they run it back with Rizzi?
We're definitely not the worst landing spot, but we ARE a very mixed bag. I think the Saints organization has a good reputation for being pretty buttoned up in general where you couldn't say that about Las Vegas, NYJ and maybe Jax. I think the Pats and Bears jobs are both much better options having just drafted decent young QBs and having a ton of cap space to shape the team. The Raiders also have a ton of cap space and a really really good TE, but no QB.
The Saints have been loyal and patient to a fault, but that won't be seen as a negative to a coach. What will be seen as a negative is the lack of immediately available resources to make the roster the way they want it to be and that might be a turn off for some coaches that want to put their stamp on things quickly and not coach another guys left over team.
As far as Rizzi goes, if he is allowed to interview with the Jets I think he'll make a good impression there. He may not be the flashiest choice for them, but he's tough and the fans will love him and that might be enough. So I don't think we'll get to the point of having to run it back. There are enough candidates that know this team that one of them will come here and Rizzi will be on his way.