Actually, you're wrong. The DMCA was used as a premise for takedown on the basis of purported copyright infringement. I was pointing out that such a claim, however spurious you may think it is, would likely hinge on the claimant's exclusive rights in the compilation. I can't see any other theory that would support a DMCA takedown. Again, it's irrelevant to the question whether or not you think the claim had merit. Also, the question conflated trademark with copyright.
That's all well and good if this case ever goes to court, which it almost certainly won't. Most service providers will pull down any content that's included in a takedown notice. The alternative is for each service provider to make a legal decision about each takedown notice. That won't happen. If they don't respond in good time, they risk their safe harbor status.
It's smarter (under the seriously f'ed up DCMA system) to pull it down and wait for a counter notice, at which point they put it back up, and they aren't responsible any more — the person filing the counter notice is responsible because a false counter claim counts as perjury.
At that point, the original issuer of the takedown notice can sue.
Of course, it almost never gets that far when the takedown notices are spurious, as in this case. That's what makes the DMCA such a bad idea — it's a perfect tool for censorship.
> I was pointing out that such a claim, however spurious you may think it is, would likely hinge on the claimant's exclusive rights in the compilation.
I don't think it is that simple. Yes, you may claim a copyright on whole compilation. But you could as well claim a copyright on some parts within that compilation. Since the compilation is a single document, claiming copyright on some part of it would also be sufficient for a takedown.