Particularly for a baseball team, where the park is located is extremely important, and the wrong choice can be a very major problem. We've seen this happen with the Rays over in Florida. where the location of the park has proven to be a severe hindrance to the team's attendance, and thus fan-base. The current location can be reached in less than half an hour from just about anywhere inside the 101 (and quite a few locations outside), which would certainly not be the case if they moved to some of the more peripheral sites suggested such as over at the intersection of the 202 and 101 freeways in Mesa.
Some may point to the success of the Cardinals out in the West Valley, but the NFL is different, with less than a dozen home-games a year, even including the post-season, and most of those on Sundays. Particularly with the team opting to start weekday games at 6:40pm, any non-central location will become increasingly difficult for a significant number of working families. Right now, traffic isn't too bad coming into downtown around 6pm, because the bulk of the volume is going the other direction. But as soon as you get past downtown, say, and continue to head East, you'll quickly find yourself developing a solid relationship with the car in front's bumper.
Moving somewhere else would effectively mean starting afresh in terms of building up the environment and atmosphere around the stadium. While that offers some benefits, we've seen way too many proposed commercial developments implode or turn into ghost-towns. Tempe Town Lake. CityNorth. Even the various developments nearby downtown have not exactly taken off: you just need to wander about the ghost-town which is largely the Arizona Center, to see the potential for failure. If the people involved didn't get it right before, whose to say it will be any better this time round?
At least in the current spot, there are things like the light rail, which offer a conduit to bring people in. And the current location is entirely fixable. The obvious step would be to get rid of some of the car-parks which can be found occupying the nearest real-estate to the stadium on all four sides. Take down, say, the Jefferson Street Garage, and you'd have plenty of space to develop an entire bevy of restaurants, etc. that would make the area more of a destination. Though the problem would be finding a way to make it work on the 284 nights of the year when there isn't a Diamondbacks' home game taking place, that would apply anywhere else too.
Getting the money to build a new park somewhere else would be difficult, especially in the post-financial crash world. Although the Braves have one under construction at this point, the only major-league stadium to have opened since then was the Marlins' one, which managed to squeeze its funding in February 2008, just before everything melted down. The apparently unending spigot of public money for such private enterprise appears to have been restricted significantly, considering that between 1999-2010, no less than fourteen new major-league ballparks were opened, with just one (AT&T Park) privately financed.
It's also worth considering costs would be significantly higher than than the $354 million for Chase Field. A good estimate would be for that Marlins Park, a retractable-roof stadium like the one here. Adjusted for inflation to 2016 dollars, it cost $653 million, close to twice the price for Chase at the time - that's also in line with the cost of the Atlanta facility ($622 million). Funding for the Braves' park is being split between the team ($230 million) and Cobb County ($392 million), the latter raising almost all of the funds (over 90%) not through taxation - as was the case for Chase - but through issuing bonds. Would any suburb be able to put together a similar deal?
While split, the public in Georgia were generally in favor of the package, given the lack of tax dollars. However, it's worth noting a significant difference is that the Braves were moving at the end of their least: the Diamondbacks still have more than a decade left on theirs, and from what I've been hearing, that looks to be having a big impact on opinion. Many people also remember the dubious shenanigans which took place before the funding of Bank One Ballpark [if you aren't, I thoroughly recommend Big League, Big Time for an in-depth recounting of the way it was pushed through, without any chance for the public to vote].
I still tend to the view that the request to explore other options from the team, was more a negotiating ploy than the first step towards an actual relocation. Without seeing the lease, I can't be sure, but I suspect that it was likely written to be watertight, by the best lawyer the Maricopa County Stadium Board could buy. and breaking it would not be easy, and would cost the team a lot of goodwill. I'm fairly sure it does not promise the Diamondbacks the "state of the art" facility which Derrick Hall demanded in his Chase Field press conference, even if this term is used frequently in the team's letter to the board.
Mind you, if the team wins the World Series this season, I think they may well be able to ask for a new park, just about anywhere - and even I would happily chip in a few dollars for that!