The productivity gains are not necessarily such that they contribute to growth -- at least, not they way we are used to measure it.
It seems that two of the big effects of the Internet is removing rentiers and middlemen, and lowering costs across the board, the first because, well, connecting people, the second because of multiplicative efficiency increases.
This enables smaller organizations to capture a bigger part of a smaller pie. This is most visible in strictly informational sectors, such as free software, music and publishing, but I don't think it's constrained to that but rather the effect of adding "intelligence". The end game of a company like AirBnB for instance, is to replace (all) hotels, for a fraction of the turnover (but higher profit margin). The same could be true of self driving cars -- we believe it will let us make do with fewer cars, because they are easier to share. Indeed, Google is perhaps the foremost example, incessantly cutting away at the advertising sector (along with related industries), filling the same purpose but better and an order of magnitude cheaper.
This is not growth, but degrowth. The thing is, it is still good and valuable -- it's just that it's not reflected in GDP.
I would say that this has been going on for a few decades, which is why we've seen the rise of the financial sector, which seems, to me, to be a kind of "manufactured growth" -- all of our politics and policies expect, indeed, demands growth in order to work. So when it slows, or rather changes it's nature if you will, we invent fake growth.
In summary, I agree with some of the "pessimists" in the article, but I'm optimistic and hopeful about it. I think we can really live abundant.