I don't believe that local or regional governments ought to be underwriting the entire cost of a sports complex for the primary benefit of a single team. However, even a large privately-funded project, whether it's a stadium, industrial park, shopping center, or whatever, usually incurs some kind of financial participation from the local governmental agency.
The Sacramento region, where I live, is a fairly large media market (about #20 nationally in size) with a single major-league sports team, the Kings. The previous owners had an increasingly strained relationship with the city and the fans, had no serious interest in building a new arena, and just wanted to sell the team to buyers in Seattle. Local civic and business leaders came together and put together an ownership group committed to keeping the team here, and outbid the other contenders. The NBA required a new arena to be built as a condition of approving the sale to the local buyers.
The team and the city chose a downtown location and agreed to split the costs roughly equally. There were several court challenges to the city's method of financing its share, but the court cases were all dismissed as the plaintiffs were unable to prove their claims of backroom deals, financial disaster, and gridlocked traffic. Construction started about a year ago and the new arena is more than halfway complete. Opposition seems to have receded into the shadows. The arena project has sparked a surge of investment in the surrounding area, including a new hotel, office space, and revitalizing older structures to bring in additional business. The arena itself won't be the biggest and flashiest in the NBA, but that's OK. It works for us.
Major league teams bring a certain recognition to a city which might not exist without the team. Sacramento is like that. We don't have a Fortune 500 company headquartered here. The biggest employer by far is the state government. We're less than 100 miles from San Francisco and there's a long, complex history of rivalries between the two cities which has lasted for a century and a half. There's a huge fan base in Sacramento for Bay Area teams, especially the 49ers and the Giants. But the Kings are ours (yeah, I know, they've been a sub-.500 team for several seasons). And without that public-private partnership arrangement to build the arena, they'd be probably playing in Seattle.