The traffic lights in Chicago are likely made by a different company than the traffic lights in New York City (I dunno for sure... but it'd be a separate arrangement nonetheless). Those are two grossly different customers.
USA has a federal society: that means we have mini-governments across the country with their own laws and regulations. In an extreme case, an Home-owners association can own traffic lights and operate roads. (My own HOA for example owns the traffic lights in my area, and hires cops from the county to patrol our roads. They're county-cops contracted out to the Homeowner's Association for security in the neighborhood)
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Same thing with voting booths, and other services. They're made by private companies in the USA, to serve the many different local governments that exist.
And local governments interact with local neighborhoods / homeowners associations, depending on local regulations of course.
The underlying argument you're making comes down to this idea that capitalist competition is the most efficient way to produce results, but it's obviously incorrect in this case. Are we really arguing that having a bunch of different companies developing the same software a bunch of times for different governments is the most efficient way to do this?
That's an assumption that isn't always true.
The company could have developed them with their own funds, then sell the product to the government.
But government pays them to develop the code. For many such companies government is the only customer (whom else are you going to sell the traffic light software to?).