No, it’s the site owner choosing to outsource their decisions about gatekeeping a private site to Google. Google isn’t censoring you via CAPTCHA, the site owner is.
That is only true as far as the site owner knows of and understands the consequences of their actions. I would be extremely surprised if more than 10% of their users understand this. Whether they care is a whole other matter, but this is very likely ignorance rather than malice on the part of site owners.
I can understand why site owners resort to such services. They need a a strong CAPTCHA system. The problem is really Google for abusing it.
Lol sure, and “guns don’t kill people; people kill people”