Perhaps the Lyft statement (Lyft left too) says it the clearest:
"The rules passed by City Council don't allow true ridesharing to operate. Instead, they make it harder for part-time drivers, the heart of Lyft's peer-to-peer model, to get on the road and harder for passengers to get a ride."
If you read more than just one article on the issue, you can see that it's much more nuanced than how you represented it. They left because they both have background checks in place and that adding fingerprinting adds an additional bureaucratic burden that has poor predictive power, places an undue burden on drivers who want to engage part-time and disproportionally prevents minority communities from participating. Background checks add nothing of value and if either company were to validate Austin's approach by acquiescing, they would pave the way for other cities to follow that model.
Opposing measure on principle because it adds costs and friction without adding value is not equivalent to the statement you made. Every business is right to oppose regulations that increase costs, especially those regulations of highly dubious value. Opposing such measures is not evidence of a business model not being valid.
Thus, I do agree with you that that the only thing that would change is the supply and demand curves, but many of these business models seem to depend on the very idea of there being a large enough supply to enable prices to be lower than alternatives.