They might not love it the same way, certainly. I'd bet that many very great athletes don't "love" the game the same that the fans do. In Ball Four, pitcher Jim Bouton wrote about having to pretend to be super angry after a loss to satisfy the media. He wanted to play well and did everything he could to win, but he just didn't relate to the way sports writers and watchers relate to the sport.
He certainly wasn't a hall of famer, but he had more sports success (world champion) than probably anyone on the field today will. I'm sure attitudes vary among elite athletes, but I think as spectators, we overromanticize the stuff about hating to lose, love of the game, and other symbolism. John Wooden didn't win all of those championships because he taught his players how to tie their shoes; he won because he paid the best players in the country to come play for him.
I get the desire for some acknowledgement of the fans after a game and I get mourning a cherished tradition fading away. Those are fair feelings, but I think there's a lot of reading too much into this in terms of it being related to the poor play.
The team lost because they aren't that good and they're poorly coached. They make timely, dumb penalties that kill them. They make mistakes that turn the ball over. They don't block well. That's not a lack of unity that's solved by singing songs together, it's just stupid plays and poor execution (ie., a mediocre team poorly coached).
Especially when the team is losing, we often focus on things that are tangential to success on the field. It all reminds me a little bit of the Iverson "we talking 'bout practice" rant. Rage on surly, but the team could have concluded with a perfectly choreographed Haka to the Eyes and they would still be a mediocre team that is poorly coached.