In Arizona, it's because of taxes and the citizens' utter loathing thereof. Not to mention that building a truly competent metro system in greater Phoenix would take at least a decade and would badly disrupt the metro area (see Boston's "Big Dig," but on a much larger scale). So most people (not me) feel like it'd be a waste of time, money, and effort.
Very similar to some sketches I have made over the past 30-some years (although the central station would be closer to downtown proper and some of the routing would be different). Mine were based on how the Boston and Washington metro networks are laid out but they follow the same hub-and-spoke pattern.
Anyone who's ever used a good metro system loves the efficiency and convenience. I've used metro systems in Boston, Washington, D.C., Mexico City, and Moscow, and all of them were enormous time-savers (although the Moscow system was so overloaded that it wasn't nearly as pleasant as the other three to use). Particularly in Boston - I lived in New Hampshire and simply drove to the metro station just over the border from Nashua, parked for the day, and took the train in to my job, or to Fenway for a Sox game, or downtown for sightseeing. My wife would take it every week to go into the city with my then-toddler daughters to see the sights. It was fantastic.
Unfortunately, not here. The most basic problem is that the ground is simply too hard to easily tunnel a metro system - it'd cost a lot of money to go underground. The alternate option would be to do an elevated train/monorail system (cue Simpsons gag here), which, if solar powered, could be an interesting green alternative - but I don't know whether any metro cities have an appetite to have something like that running through their towns.
Football stadiums are significantly different than ice hockey arenas. We have two convertible-roof stadiums already in Arizona - Chase Field and State Farm Stadium - and both have massive air-conditioning systems to help keep the fans relatively cool. But the coolest they get with the roofs open is around 78-80 degrees Fahrenheit; in contrast, a hockey arena needs to be in the low 60s to keep the ice at a manageable temperature and consistency for gameplay. We already get complaints from teams around the league that we have some of the worst ice in the NHL because we have problems keeping the arena cool enough (although I'm sure our ice plant is partially to blame for that as well).
It's an interesting idea if someone could get it to work, but we don't have Qatar's budget...