These qualities are present because they were superior to the previous market provider, in order to compete in a free market your product or service has to be superior in some manner. Competitive market forces like safety, efficieny (both price and liquidity) are what makes Uber successful.
No laws dictated these qualities, no legislation demanded this level of safety. These practices were conceived via market forces, and safety was a by product.
The entire point of Uber is that the taxi industry is not a free market, therefore the things necessary to compete in a theoretical "free market" are not relevant and can't be assumed to be why Uber is successful.
Uber uses roads that were designed to comply with government safety regulations and their drivers drive cars designed to comply with a different set of government safety regulations. Those regulations exist because experience has shown that they need to.
Unless I'm wrong you're arguing that Uber is safe and efficient because market forces made it so. It's efficient because of technological advances and it's safe literally to the extent required by law.
Because they've opted out of the regulations their competitors are subject to Uber could offer a a more dangerous and less efficient service than current taxi companies and still "compete".
Safety precautions are in fact codified by law. Uber is in the business of breaking the law and ignoring all regulations. The fact that they have a substitute that follows the spirit of the regulations if not the letter and provides safety is the only reason governments have chosen to allow Uber to exist. The safety precautions Uber has ARE required by regulation, though implicitly; they act as a substitute which makes Uber's lawbreaking far less egregious.
No laws dictated these qualities, no legislation demanded this level of safety. These practices were conceived via market forces, and safety was a by product.