The article claims that the data "should be broadly representative of the U.S. population".
Did they filter out all government IPs to reach that conclusion (assuming IPs are even icluded)?

My first guess would be that a large proportion of page hits in that dataset would be accesses from within government organisations. This would then explain why the summary reminds of other statistics dating 5+ years back. Government agencies tend to be slow with hardware and software updates, after all (and not always for bad reasons). Btw.: Did one or several big government organisations recently switch to Windows 8.1?

Did anyone look at the raw data, to verify or correct their conclusions?

The only time where the author acqnowledges a possible "in-house effect" comes at the very bottom relating to Blackberries, but that thought didn't seem to influence the rest of the analysis.

There's an update to the article now that explains that the most popular sites in the data set are consumer facing websites, so he argues that it is representative. I guess others raised similar questions in the comments.

If we accept that, the question then is still, representative for what?
Obviously for the web audience of a government institution. Since those sites tend to deal with "serious" topics, people may be inclined to visit them from work, of from their desktop computer at home, instead of with their smartphones.
But are those figures also representative for a general web audience, or even for an entertainment-seeking web audience?

Still interesting to see and compare those figures, of course.
Its always good to realize that your own figures may also not be representative for every offering and its audience.

Well the update did mention that the most popular sites in the collection were the Internal Revenue Service (with the "Where's my refund?" page) and the National Weather Service, with its popular forecast pages. I would assume that those have a pretty broad appeal to people living in the US, so we may be looking at a more diverse spectrum than expected when thinking about "government websites". So it may be, as the author claims, that this is broadly representative of the U.S. population.

While all those sites may have broad audiences, just those two top examples will certainly have very different browser stats.
People are much more likely to fetch a weather forecast from their mobile than an IRS refund query page.

I didn't download any of the data, but the categories of files offered suggest that browser stats are only available globally, and not by site.
Being able to compare access methods between those diverse topics would be most interesting.

(with the "Where's my refund?" page)

This page/search is not going to be representative of the general population.

Generally, the more money people make, the less they tend to get refunded back, at least proportionate to their income and therefore not a topic of concern. Then there are those who have to pay, they are not doing this search.

It is probably pretty safe to say that the majority of these searches were done on older machines by people on the lower rungs of the economic ladder as they are the ones most in need of the refund.