Conflict of interest means: "A conflict of interest is a situation where a person or organization’s personal interests—financial, personal, or professional—compromise their professional judgment, duties, or obligations."
If you're romantically/sexually involved with the coach while reporting on the NFL, for any story that puts info out the Pats might not want out there your professional duties to report that now have a negative effect on someone you have a personal relationship with. It's a clear conflict of interest.
What if NE was trying to move up for an OT? It would benefit them that it not be reported they're doing that so no one moves up further than them. Russini reporting that interest would hurt a guy she has a personal relationship with. You can't have that. Maybe she would report the story without bias, but the fact it's even a situation to consider is why it's inappropriate.
A guy and a girl dating CAN in theory work together in any business. There's no inherent reason why a boyfriend can't give an honest work performance evaluation of his girlfriend. But we all know the possibility for bias is so obvious that it's a dumb situation to allow. You can't have a national NFL reporter banging the head coach. The chance of bias is so obvious.