Yes, the author is debating a straw man. The argument that the government is understating inflation is a pretty simple one.
1) The government is heavily incentivized to understate inflation in order to limit its expenditures for social security and TIPS, both of which are indexed to inflation. Social security in particular would explode in costs if, say, the official rate had been 1% higher on average over the past 30 years since the inflation measure got significantly changed during the late 80s and early 90s. In fact, if you look back at the discussion surrounding those changes, bringing down the costs of social security was an explicit reason for the redesign.
2) Complexity and obscurity. None of the input data is released, and there's this thing called 'hedonic quality adjustments' which is the umbrella beneath which economists decide how much to lower inflation when the quality of goods gets a little bit better. The FED economists point out that a car from 2018 is better than a car from 2012 and they adjust inflation downwards in order to account for the increase in value that consumers are getting in their car. Consumers counter by pointing out that they still have to spend the actual money...
Analysis) With transparency and a lack of incentive to cheat (see 1) hedonic quality adjustments would be an allowable modification, but with opacity and a strong incentive to cheat, they pretty much guarantee that inflation is fudged downwards. The debates around changing the inflation measure have focused heavily on reducing social security payments. In short, any time you have excessive complexity, lack of transparency, and an incentive to cheat, you can expect cheating.
Why should we care? First and foremost, Anyone who has a relative collecting social security would probably be upset to hear that grandma is getting about 60% of what she should be getting, given the promises made and the value of the money she contributed.
Second, and the big point that the article missed entirely, is that real gdp per capita HAS been declining, reflecting that decrease in standard of living that one would expect to see if inflation were understated.