He definitely wasn't "pushed out" in the traditional sense (see: Gerad Parker, Charley Molnar, etc.) where you tell someone 'thanks, but go find another job.' I'm shocked to here the LSU guy saying that, because it isn't even cope. It's just weird.
The alleged story from the time of the hiring is that the contract extension wasn't fully executed by LSU's board in a timely manner because they were being lazy // it was over the holidays. While in limbo, ND made overtures and he wanted to gtfo from LSU. Why? Well for starters, he has a history with Kelly where Kelly did not treat him well from 2014 to 2016. He was OC in 2014, then Kelly wanted Sanford as a QB coach and had to give him OC title to get him. So he made Denbrock AHC, and despite Denbrock being the party most responsible for the 2015 success it wasn't Denbrock getting the lion's share of the credit + compensation. And then Denbrock was not treated well during the 2016 season. Then comes the reunion, where he gift's Kelly an elite offense + Heisman winning QB... and Kelly tries to reverse-Sanford him. Give Sloan the credit + a promotion to keep him, especially because Sloan was the primary recruited on Bryce Underwood and Kelly didn't want to lose Underwood. So there was not a great working relationship with Kelly and Denbrock after '23, but Denbrock (who is old) was content to take the pay raise and gut it out a few more years and then retire. When the ND offer came, he snatched it up. LSU tried to counter and match ND and he told them to fuck off.
A team does not counter offer and try to keep you if they are "pushing you out."