Technically you're right. You can run a website that gives users a service at no cost without violating privacy.
Pragmatically, however, it's very rare and 'free' correlates very closely with privacy violating business models. Talking about how free services violate privacy is fine when 99% of those services do exactly that. Those businesses shouldn't get a free ride on harvesting and selling personal information just because a handful of other businesses manage to be better.
Your argument, even if considered valid only cover one part of the problem with the title. Many paid services also violate users privacy. So no, the correlation of "free" and "violates privacy" is not that strong.
There's plenty of services that try and operate under the 'freemium' model, but they're often pretty restrictive, and website hosting is often bottom of the barrel scraping lvl when it comes to that.
Pragmatically, however, it's very rare and 'free' correlates very closely with privacy violating business models. Talking about how free services violate privacy is fine when 99% of those services do exactly that. Those businesses shouldn't get a free ride on harvesting and selling personal information just because a handful of other businesses manage to be better.