To build off ejpsan's last comment, Chicago winning the bid would probably be a bad thing for the cost of the games in the short term. Rio's experience seems to have been a key piece in the pattern that's leading to reform. Whereas spiraling costs would be less compelling in a Chicago games- just another rich city in a rich country that can afford to splurge, even if they're not happy about it. I would say where we are IOTL people are starting to deal with the reality that costs are out of control. ITTL, it might take another cycle or two for people to come around to the inevitable need to reform.
On another note, it's worth thinking about how this might affect Obama's presidency. I could be mis-remembering, but I recall the lost bid coming across as kind of a public defeat for him; in fact, I remembering it being his first public defeat worthy of occupying a news cycle. Without it, it's not unlikely that he makes it to 2010 without any chinks in his armor. Now, it's an Olympic games, not a piece of legislation, so it's possible to overstate the effect of that. But on the other hand, we've got plenty of evidence in our current situation of how powerful a tool perception can be, even on members of congress.
So there is perhaps a chance that fewer people line up in opposition to the legislative priorities of early 2010, most notably the ACA. It could be an easier passage, it could even be marginally more progressive. I think if someone wanted to include that argument in a TL it would be as valid as most suppositions that are found to be reasonably acceptable on the board.
Conversely, drawing even more attention to Chicago in 2016 probably leads to political headaches for Obama and probably the Democrats. Even if the reality of Chicago is different from current perceptions spread by certain swaths of the media, more scrutiny is more likely to magnify those perceptions than dispel them. And every dollar spent on the games- even if price-gouging doesn't happen (which omg of course it will)- will be decried as wasteful graft by the Republicans. Every incident of crime in Cook County deemed worthy of making the police blotter in the local section of the paper will become a national story for five minutes. It's not out of the question that some of our more outrageous editorialists on the right start making comparisons to Obama and Hitler and the 1936 games. And if the Democratic dynamics are the same- Sanders v Clinton- there's no reason why the left wouldn't pile on at least about the boondoggle. Protests in poorer neighborhoods about how they can't get basic services while money gets funneled into stadiums.
But in the end I can't imagine the games themselves not being pulled off. It'll be a net positive victory lap for the president. And would another two-week tempest in the 2016 election cycle REALLY be all that noticeable? Maybe Sanders does a little bit better.
Brazil, of course, saves a lot of money and goes along with its economic woes in a much more low-profile way.