I may pay more or less attention to some sports at BYU at any one time based on how they're doing. I think that's human nature in a busy world. But cheering more for a different team? Never. This is who I grew up with and it's a part of me. From my dad taking me to games back in the 70s as a young kid and through all those years to today. They could be bad on the field for years but it could never change what's in my heart.
Sure, those factors you mention that can connect you to another school (relatives playing a sport there, your own kids attending there, being hired for a job at that school, etc.) could definitely come into play somehow and lead to me following another team for a while. But it would still be different. I had a good friend who years ago got a job at the UofU. He went to BYU, was a huge BYU fan for most of his life. Now as part of his job he was expected to go to the Utah games, which he did wearing a red sweater, which he said he'd still have a BYU t-shirt on underneath whenever they played BYU.