Large businesses do similarly dumb things with digital products and services. There's nothing I see here that's specific to government, except that there's a publicly published report detailing expenditures and problems, which you generally wouldn't see in business.
And, frankly, without reading the actual report (which the OP doesn't link to) it's hard to know if you're getting the whole story here. The Washington Examiner is a right wing publication, so it's not hard to see where they might selectively present details to paint this in a negative light. Not saying that they are distorting things, but absent a review of the original document, it's something to keep in mind.
It looks like the document is included in the article in a PDF viewer. Sometimes those don't show up because of ad-blockers.
The Examiner does tend to cherry-pick, and it's definitely worth reviewing the document. I searched the doc for "facebook" and it looks to me like the broader goal was engagement, not "likes" as an end-all. They were working off the assumption that more likes meant more engagement, and during that campaign, people started to realize and point out that it wasn't happening. And to that end it appears to be a waste of $630k.
But it's not the case that they just blindly wanted more likes because it feels good. (The article doesn't say that, but a lot of people think about "likes" that way). Facebook and Twitter are legit ways to reach international audiences, for example, to disseminate propaganda / information (depending on how you lean).
And, frankly, without reading the actual report (which the OP doesn't link to) it's hard to know if you're getting the whole story here. The Washington Examiner is a right wing publication, so it's not hard to see where they might selectively present details to paint this in a negative light. Not saying that they are distorting things, but absent a review of the original document, it's something to keep in mind.