It would be very, very unusual to hire Rizzi as a HC, for reasons discussed to death on this forum.
The side issue is what to do about Loomis as GM. At the end of the season a head coach must be hired, meaning that Loomis would normally be the guy to orchestrate that process. I have no idea what would happen were Loomis to be fired, or even “promoted.” But I know this: that the hiring process should have already started (with apologies for the Paytonspeak). So, if you want Loomis to be fired or “promoted,” there’s a problem because the Saints would need someone else to get the head coaching search arranged, and in earnest immediately after this season mercifully ends.