I think that comes down to what their reasons were for taking it down. My guess is that when editorial higher-ups read the piece and heard the context they decided it wasn’t a piece they wanted to make a stand on.
To be more specific, if this situation is the kind of thing that’s likely to blow up into a larger story that people start talking about in other publications, then Esquire would want to make sure that the piece itself was something worth defending. After all, their name would be associated with the piece in every piece of discussion that happened afterwards.
My guess is that once they’d had a look at it they decided it wasn’t a piece of particular substantive content that they’d want to fight for and so just took it down.
From what I can tell, a fair few things like this happen in media circles. Publications produce a lot of pieces of substantial journalism and will go to the wall to fight for them.
But honestly much more often, they publish some puff piece or slightly ill-considered opinion piece that they’re not particularly proud of and it lands poorly or causes a big stink. Then they look at the thing and try and decide if it was worth publishing in the first place, and whether it’s worth dealing with the negative PR for and if it is isn’t (because the article is bad, they’re in the wrong, it’s an ad hominem attack from a writer with a grudge, or just because it’s a bit tacky and doesn’t really fit with their stated brand values in some way) they just take it down.
Personally, I only thought it needed to be clearer that it was satire and once the new disclaimer on the top was put in I was quite content. So I was a bit surprised when it subsequently went down. ¯\_(ツ)_/¯