re: Saints not signing off on Nielsen leaving

The Saints don't need LSU's money for any buyout (Do they even have buyout clauses in the NFL? Never heard of one. Teams are sometimes compensated, however, when one hires a head coach away from a competitor).
So this probably isn't about money or buying out Nielsen's contract.

It may be as simple as somebody jumping the gun on the announcement. Did LSU or Orgeron announce Nielsen had been hired? Did the Saints? Neither.
The "scoop" came from Jane Slater of NFL Network, the same person who later tweeted that -- hold the phone -- there was a hangup.

The NFL has rules about when NFL teams can hire away coaches from playoff teams. (There's a waiting period following the team's elimination from the playoffs). I'm not sure if colleges have to abide by that rule. Sorta doubt it.

Another question is why would Nielsen have agreed to a clause in his contract prohibiting him from taking a college job? Doesn't make sense.
It's happened, even recently. Didn't Bielema leave the Giants for Illinois? Kentucky also hired its new offensive coordinator from the Rams.
Nielsen is doing nothing unique here.
Coaches can leave for other jobs. Period.

As we saw the drama unfold Monday night, the optics are terrible for the Saints, of course. Why would they want to piss off a major portion of their home fan base?

I'd be willing to bet this was a procedural screw-up.
To me, all signs seem to point to Jane Slater's "scoop." She spent a few days in New Orleans before the Bucs playoff game. She might have been told something and then jumped the gun before the Saints and LSU (and Nielsen himself) were ready to officially confirm it.

Fake news, in other words (for now). But nothing the two teams can't resolve Tuesday.