I can answer my own question (in the title here): yes, they do. It’s annoying, really, but they do.
It turns out that promoting something with “special offer” language still works for some people. You don’t get me, I’m afraid. I always assume that’s just hype. And my advice to you is that they shouldn’t get you either.
“Special offer” means to me that they couldn’t think of anything else to put in a headline.
But then, what would I know? I’m the one who once decided to go to print with direct mail pieces offering something for $100 instead of $99.95. “That $99.95 baloney is insulting,” I said, to the direct mail expert, back in the heyday of direct mail marketing. “Let’s call it $100 so they know we’re up front.” Bad idea.
And what happened, of course, is that the response rate went down to less than a third. Live and learn.
So, keeping that in mind, if you look at the search results shown here, “special offer” is up there 29.9 million times (no, not 30 million, just 29.9 million). So it probably works.
But not for me.