I don't think the article was claiming that this is an innovation outside of the capitalist box - just that it's a form of capitalism that's different from the way it's been practiced in past decades.
The article does relatively little to justify that though. It's not as if companies being privately owned before they go public (or private equity buyouts of public companies) is a new thing. The time-to-IPO is the only non-anecdotal difference in ownership structures it notes, and that's easily explained away by market conditions.
Indeed the first example it cites - Uber - is arguably an example of the opposite to the trend it purports to spot... it's a market traditionally dominated by plucky privately held small companies that's becoming centralised under a large company or two that probably will go public. Even if Uber (and Lyft and a couple of others) doesn't IPO they still are and will be owned and managed in a way that resembles a public company more than your neighbourhood cab firm.
I'd give the article more props for identifying interesting but not really connected phenomena in new companies if it didn't include lines like "whereas nobody is sure who owns public companies, startups go to great lengths to define who owns what". The idea that startup share options are more transparent than employee shareholdings in publicly held companies traded on liquid markets is dangerously misleading nonsense that looks really out of place in a publication aimed at the financially literate