re: Reasons I think House is getting fired

He does. The key is how he separates from LSU before he leaves for the NFL.

House is not going to terminate his own employment without cause, as that term is defined in the linked contract, and then leave for the NFL. You are correct that this set of facts would relieve LSU of the buyout obligation. But despite what you read about House here, he is not dumb enough to do LSU that favor. There is no universe in which “House takes an NFL job without being fired from LSU,” and never has been.

He is instead going to be given time by Kelly to find a new job in the NFL. Once House agrees to terms with his new NFL team, LSU through Kelly will terminate House without cause, as that term is defined in the linked contract.

That order of operations allows House to go to the NFL, collect an NFL paycheck, and get the remainder of his LSU DC pay from LSU.