It seems quite likely that this is a mistake in Wizzair's wording. Indeed, they make pretty much the opposite statement here:
If you hold non-EU travel documents, you need to present yourself at
the check-in counter to have your documentation checked and your
boarding card stamped.
Credit goes to ptityeti's answer to another question for leading me to that source.
So, these two quotes are contradictory. One says EU/EEA citizens have to present themselves at the check-in desk to provide their documents, the other one says non-EU citizens have to do so. If we were to take both these quotes as accurate, the conclusion would be that everybody has to do it. In that case, they obviously wouldn't have to mention EU/EEA at all, they could just state plainly that all passengers have to do it.
Therefore, it seems to me that they have made a mistake in the text you are quoting in the OP, and they meant to say basically the same thing as what is quoted in this answer. However, when official sources contradict themselves in this manner, it is certainly recommended to contact them directly to ask for clarification. You have already done this. As you have suggested you will, please do share the response with us.
However, I would just go there anyway
If you don't get a response from Wizzair, I would just go to the check-in desk regardless. The company has been utterly unclear regarding this. And, since they have now said officially in writing that you are required to go to the check-in desk (and may be denied boarding if you don't), I would just do it. It may take 20 minutes, but it's better than risking being denied boarding, IMO. Yeah, it's probably very unlikely that they would deny you boarding, but in the unlikely event that they did it might be legal for them. After all, they did tell you in writing to go to the check-in desk. At the check-in desk you can also get a clarification from the staff there, and then post that clarification as an answer here on TSE. :)
Side note: Technically, it could be possible that it's in fact the quote quoted in this answer that is inaccurate, but that would make much less sense. Why would EU/EEA passengers be subject to more scrutiny on Wizzair's flights when the vast majority of Wizzair's destinations are within the EU/EEA? And why would all other nationalities not be subject to the rule? It would seem like a bizarre practice. Again though, you might want to ask Wizzair directly.