Sort of, but not really.
Here's another article by John Brice. It makes clear that BK did not ask for a raise. Here's the key statement from the article:
Kelly and others close to him emphasized that he had not had his representatives ask for any salary increase from Notre Dame and Swarbrick.
So he wasn't outright looking for a raise. However, the next paragraph says he was likely looking for an extension to the contract he had signed just one year prior. His people also indicated BK wanted to upgrade ND football's nutrition program and hire a dedicated person to oversee that program. Swarbrick acknowledged this was true and made clear that he had agreed to all of BK's demands with only a minor quibble on the timeline.
All of this happened within the last few weeks of the 2021 season. So, although BK said he wanted improvements, in my opinion, the urgency of his demands leading up to his LSU announcement was manufactured to make BK look better. Here's what Brice says in his article:
A combative urgency from Kelly’s camp had replaced what appeared initially to be a friendly, are-we-on-the-same-page-here-? dialogue.
“And if you look at what was being asked for,” said a Notre Dame source, “we were pretty much in agreement on everything. There was nothing that Jack said no to.”
So it looked like BK's camp was saying there was a sudden urgency, even though Swarbrick and ND had already agreed to everything he wanted.
Ultimately, this is what I think happened. BK always had one foot out the door, but he was willing to stay at ND as long as he had no better options. So he kept his eyes and ears open while also trying to appear to be happy where he was and wasn't looking to leave. Over the several prior years, there were rumors that his agent often floated BK's name during big-time coaching searches, and in hindsight, it's likely those rumors were true, and that it wasn't just a ploy to get ND to pay BK more money - BK really was looking for greener pastures.
I guess my main gripe with the notion that BK left ND mainly because he wanted more money is that I don't think that's true. ND says he didn't ask for more money (but probably wanted a longer contract), but I contend, even if he had been up front with ND and ND matched LSU's offer, BK still would have preferred to go to LSU.
I guess we'll never know for sure, but I think most of the evidence points that way.