I think the better question is whether a non-LDS person would take the job. I don't think any such person should take the job. It's impossible to understate the degree to which the LDS dominates culture, business, entertainment, everything in SLC/Provo, and it's turned up to 11 at BYU, and then that is turned up to 1,000,001 in the football program. It would be really hard, if not impossible, for a non-LDS person to have the credibility he needed to corral NIL $ in Utah, develop the relationships with business leaders needed to sustain a program, and basically just to be respected.
And to several posters' points above, I know a lot of LDS plus-size folks. Utah in general is healthier than some other parts of the country, largely due to its outdoor activities and the larger percentage of outdoor recreational athletes, but that's not an LDS thing. They're just as much victims of our obscene American diet and lifestyle as the rest of us.
(I'm not LDS, btw. Not even close. But I do know a number of them).