This (boosters having the power at UT) is such an oddly and often repeated idea.
Mack Brown got comfortable and fat and lost his edge, and then Texas hired a string of bad coaches. It's pretty simple.
I suppose the argument could be made that the boosters influenced the hiring of that string of bad coaches, but most people thought Charlie Strong would be a good coach at Texas, and a lot of folks thought Tom Herman would, too-- Texas had to outbid LSU to get Herman on campus, so it's not like Texas was going out on a limb and hiring an undesirable from out of nowhere.
I don't have much hope for Sarkisian, but it wasn't the boosters that selected him, it was the university president.
Yes, it is odd and often repeated.
Back in the "Before Times," when there were a number of Texas posters--Hooky, Burnt, and Erin come right to mind, but there were others--on the old board, there was periodic discussion of the disappointments of being a UT fan. The "Austin malaise" was discussed. And so were the "PTBs"--the powers that be. I always took that to mean the boosters.
Amongst OU fans and homer mediots, there was some commentary on the announcement that the UT starting QB was going to be Quinn Ewers rather than Hudson Card. Why was it the UT SID rather than Sarkesian who made the announcement, and why did he make it at the very end of a press conference as the mediots were practically putting their laptops away? The theory--and comments from Texas message boards were cited in support--was that it wasn't Sarkesian's decision to make, or that Sarkesian had decided to start Card and the boosters wanted it to be Ewers. For NIL purposes or something.
It seems that, when Charlie Strong was hired, there was some semi-public grumbling from big donors.
And Tom Herman seemed to be caught on the Horns of a dilemma during the "Eyes of Texas" controversy--whether to satisfy his players or satisfy the donors.
Anyway, as you say, the idea that the boosters have an excessive and damaging influence on the Texas Longhorn football program is longstanding. Is it possible that they don't have much say on hirings but have significant influence on firings?