Only if the league agreed they could use their trademarked phrase. It’s not about acknowledging the trademark, it’s about getting permission from the holder.
Well, I’ve used trademarks zillions of times in formal documentation and all I’ve ever done per guidance is add a footnote that e.g. “Teflon is a registered trademark of E. I. du Pont de Nemours and Company.” duPont probably prefers that over using the generic PTFE.
Now, an event like “Super Bowl” is different I guess in that you really can’t use the term willy nilly even with a clarification.
Bottom line, I really couldn’t give a crap about the “Big Game” - it’s about as interesting and overblown as the SOTU (TM).