The article cites a study on (USA) GDP as a metric measuring of productivity, however, GDP is a poor proxy measure of "things being produced". USA GDP includes financial services and consumer spending, so using debt to pay for things vs. producing more, can still boost GDP.
In the United States, pre-Uber around 2010, it was very hard to find cabs in certain cities and even neighborhoods in those cities. New York City had no problems, but as I had lived in Chicago and San Francisco with lower population densities compared to top tier cities in the world -- the only way to get cabs even a few miles outside the main downtown area, was to call taxi dispatch, if you knew their number -- it was even hard to look it up online -- and schedule a pick up, which usually took 30min+. The US taxi system was stalled at the time, and Uber forced it to start adapting to the smart phone era. The tragedy is that the people investing in and also running uber had their own pyramid scheme instead of pricing out the business correctly. They also didn't consider the social/gov implications -- places with good taxi systems didn't need them, as you alluded to with the UK, but they shoved into it anyways.
So Uber was an innovation, one that is trying to find its financial balance still. I'd never care to schedule another taxi pick-up for next morning, 16hrs in advance when I want to go to the airport.
But the article saying that uber innovation and having others try it on other industries is stymying productivity isn't completely correct either.