Should anybody really care about professional team sports unless you’re actually a sports bettor ?

Anyone who chooses to care, should feel free to care. Many people get quite passionate about their fave sports teams, even if they don't bet.

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What's your question about running? You've allowed your question to sit in the running category, so it should be about the category topic.

Everyone has hobbies.  Most of them are stupid.  It is polite not to call out others for the stupidity of their hobbies, lest they decide to call you out for yours.

Yes. Different people have different interests and a lot of people care about sports.

Hmmm. Answers to this question would necessarily be opinions. This would be a good question for an "Opinion and Surveys" category.

In my opinion, no. 

With some exceptions, professional athletes in teams of major league sports are playing for their particular teams because they were drafted, traded, or chose the team that offered the best contract. Before joining the team, they did not necessarily have any association with the city where their teams play home games. Their loyalty to fans in the area lasts as long as they are with the team. When they are traded or become free agents and agree to play with a team in a different city, their fan loyalty shifts to the new team. 

If, as a sports bettor, you place bets on a game-by-game basis, you care about the outcomes of particular games, not necessarily about teams themselves. You care about the well being of the players because it helps you win your bets. You might care about a particular team if, at the beginning of a season, you bet on where the team might be at the end of the season. 

Organizations such as the NFL, MLB, NBA, NHL are entertainment organizations. I'd rather support college and other amateur sports.