due to the potentially imperfect anonymization

I don't understand what you're saying above, but my point is that disclosing any info to adversaries is invasive even if the anonymization is 100% perfect. The potential imperfection makes it worse, but that's a side issue.

An example is polling. Some terrible politician X wants to know what voters think of issue Y, like "35% in favor". So she hires a polling firm to call people and ask their opinions about Y, with the result being completely anonymized and aggregated, again, like "35% in favor". What will X do with that info? Something bad, of course! We said at the beginning that they are terrible!

So do you want to cooperate with such a poll, that X commissioned to serve an evil purpose? Of course not! Or at least, I hope of course not. In that case, what do you think of software that effectively enrolls you in such a poll against your wishes?

If your private activity is being statistically reported to your adversaries, your privacy is being invaded even if there is zero PII in what the adversary gets. This is infosec 101. A quotation due to Silvio Micali is "a good disguise does not reveal the person's height". Statistically summarized information is still information, and calling it otherwise is self-serving nonsense. You want to give the adversary NO information. Anonymization is irrrelevant.