Of course they could not compete: if your competitor flaunts the law, avoids the regulator, does not pay local taxes and externalizes a whole pile of things then there is no level playing field. It would be extremely surprising if they could compete.
Good for you. That's not the case here. Taxi companies employ people, pay wage taxes, sales tax, have their vehicles inspected once per year and in general are marginal business, except for the few in the biggest cities where it is a good business. Uber only went for the easy wins, siphoned off a large chunk of the profits in return for people working without a safety net and who do not pay into the social system, which works fine until it doesn't and then society has to pick up the tab.
The regulator is the taxi industry. More local taxes are paid because more drivers exist. The rules around the playing field are in favor of existing monopolies and they haven't changed.
The existing cartel wasn't fair. Having Uber open the door has allowed smaller players into a closed market. The taxi industry is still healthy and slightly more modern because of this.
