I guess it depends on your definition of "build". If you mean in terms of property development, then yes. If you're asking if these establishments were beneficial to the city of Chicago, then no.

As you mentioned, Las Vegas is probably the most exceptional example, but there's a reason why it was so attractive to mobsters. Casinos are legitimate businesses that also offer lucrative opportunities and advantages for criminal activity. The constant flow of cash going in and out makes it easy to launder money. The Bank Secrecy Act was amended to address this but gangs have still found ways to circumvent the reporting requirements and avoid detection.

Wherever there's gambling, there's also an abundance of drinking, prostitution, and illegal drugs, none of which are conducive to good decision making. All these factors will influence a habitual gambler to gamble even more, inevitably ending up in debt and most likely turning to a loan shark for help. People who use contraband drugs and take out illegal loans are very unlikely to seek police for help, making them easier to extort from.

The Chicago Outfit had speakeasies (obviously), brothels, and leveraged legitimate gambling enterprises for off-track betting and skimming. In both Vegas and Chicago, most of the money made from legitimate enterprises was funneled back to fuel their illegal activities which ultimately made them a lot more money. The money that Vegas and Chicago received from legitimate mob businesses was offset by the cost of trying to bring the gangs to justice for bribing police officers, smashing apart labor unions, extorting local businesses, smuggling illegal goods, running protection rackets, murdering anyone who got in their way, and basically making life a living hell for law-abiding citizens.

The slightly jokey answer is Cermak road, named after the former mayor assassinated in a botched attempt against FDR...except that Chicagoans (and FDR himself) have always contended that the assassin was out for Cermak anyway as he was a reformer and shooting him in front of the President was just chutzpah. Then again, the question is whether Cermak was that much of a reformer anyway as opposed to someone who was against the Outfit or Italian organized crime. Whatever side you come down on the facts of the assassination, there's clearly a sly joke in the choice of naming that particular street that goes straight to Cicero.

I have trouble answering the question to the extent that I don't know about the history of Vegas so I don't have a point of comparison there. The other problem is that major corruption in Chicago pre-dates, or at least is synchronous with, the rise of the Outfit and mob power, so you get to question what's a criminal organization and what's just politics.

If you look at something like the Burnham plan, which clearly shaped the face of the City dramatically, it comes off as a high progressive thing and hardly the sort of project you'd associate with vice. But the at the same time, politicians realized it was an unparalleled opportunity for graft. This sort of trend would continue. Public works were basically Daley's panem et circenses, and there was a widespread notion that corruption was justified as long as the corrupt gave the City itself it's cut, hence the Picasso, but probably more importantly countless other little projects to put City work in the face of the electorate and distract from the overt organized crime ties.

Now, is that organized crime building Chicago? Not really, insomuch as I understand the mob businesses to be the prime drivers of what was going on in Vegas in terms of bringing the business in the absence of the other failing businesses. Chicago was a center of industry and transit, it just so happens that it also had a vice industry. Likewise, there's the question of whether there's anything special about the mafia-as-the-mafia, and whether these are byproducts of the culture of corruption, and, notably, to question the corruption itself as being in any way necessary.

So your safest assumption is that organized crime (in one variation or the other) and the legitimate political leadership of Chicago were cough so intertwined that distinguishing the acts of one or the other is a less relevant distinction to make, even when there aren't any specific mob goals at stake.

(Sources: Gus Russo's "The Outfit" is a great overview of the growth and development of the Chicago Outfit. James Marrimer's "Grafters and Goo-Goos" is a great history of corruption, reform, and as is his constant refrain, how you often can't decide who's which in Chicago. There's a lot of Mike Royko that touches on this, but in particular here some of the columns in the "For the Love of Mike" collection, his WBEZ interview, and "Boss.")