That doesn't really make sense, and is an inaccurate representation of what happened and why. I wasn't in Austin while this was going down, but I know a lot of folks involved in city government (and I have more harsh words to say about the city's government than most people in many areas).
Uber/Lyft is who wanted special treatment; not the other way around. Every company/individual that sells rides in Austin follows roughly the same rules (with some tweaks to suit the mode of transit): cabs, electric cabs, pedicabs, etc. They all do the same background check and fingerprinting process.
Uber/Lyft wanted to do their own thing. The city said, no, you're going to have to do it the same way all of your competitors do it. Uber/Lyft spent nearly $10 million dollars (enough to pay for hundreds of thousands of fingerprint background checks, probably a decade worth of drivers) to do an end run around city council with a ballot proposition to not be subject to the same rules as their competitors. That proposition failed; badly.
Uber/Lyft left in a huff because they would not receive special treatment.
It is a remarkable act of rewriting history to claim that Uber/Lyft are being stifled by cab companies when they're only being asked to do what even the bicycle pedicab drivers do.
If they wanted a level playing field without those requirements (i.e. cabs, e-cabs, and pedicabs, would have also been exempt from that process and instead could use a private background check instead of the city process), I might have been on their side...and, so would the pedicabbers, e-cabbers, and probably the rest of Austin. What I saw was some spoiled brats at the helm of billion dollar valuation companies trying to get special treatment and throwing a tantrum when they didn't get it.
Also: The background check that Uber/Lyft claims prevents them from being able to operate at all in Austin costs about $40. And, both companies operate in other places that require similar processes (and every other company that provides rides in Austin does the same).
They're acting like entitled spoiled brats.