The subsidizing is what made them a household name, along with the ease of use.
And the taxi firms usually won't and often legally can't use Uber's pricing algorithms which is pretty much the only secret sauce to the app, they don't want to ignore bylaws that Uber has cheerfully skirted around and they don't want their expensively-licensed drivers competing for leads on an equal footing with new Uber drivers.
I worked for a very big "Old Media" company in Australia when we rebuilt one of their key properties. Their "definition of success" was to _not accelerate the rate of decline_ of page views. Since then I've realised that these types of businesses are ran by people who are motivated differently than 'us'.
Plus any other startup or company can build such app to compete.
Any start up could build this, but most companies don't want to deal with an MVP and the bugs that come with it, or a startup that may or may not exist in 6 months, etc... Companies will over pay for enterprise grade software if it has a history of just working. This is why Uber has an advantage, software that has a track record of doing what it says it does.