Yes having any error pointed out can be frustrating.
However, the unsolicited nature of it may play an important role in this case. In order to empathize with the feelings of people who might have been irritated it is helpful to understand the academic paper publishing process.
The authors all went through a stressful process of submitting their hard work to a journal and then being evaluated by a panel of "experts". Many of them had to make changes to their papers and resubmit them in order to get published.
There's a level of understanding and expectation about how this process works. The papers aren't normally open for public comment before publication, and they don't normally get public comment after publication. They're evaluated by people in the field, and presented at conferences, and then referenced in other papers if they're influential.
Having a unknown third party with brand new possibly buggy software cast public aspersions on a paper after the fact, at a time when nothing can be done about it, is simply not helpful to the authors and is not how reviews normally happen. It's very easy to see why authors wouldn't particularly like this, even if they would use statcheck in the future.
The only real problem here was statcheck's authors publishing all the results and making a great deal of noise about it. They didn't have to do that, it is an aggressive move that was not designed to help authors, it was designed for statcheck to get attention. We have no idea how big of a problem it is, this article might have been mostly muckraking, and statcheck might be great and well liked.
Anyway, I don't think the system is broken. It is currently working better than it has worked at any time in the past, and it is continuing to improve. Statcheck might improve it more, but that remains to be seen. Other software tools already have improved it.