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It's not a cost. Google doesn't want you to protect your privacy from them. It's a punishment.

> It's very annoying that preserving privacy comes with this cost.

It doesn't necessarily have to if Google supported privacy pass like hcaptcha does. The problem is that they don't.

Why would they? Supporting privacy-preserving options is not within their business interests.

It's way cheaper than that, you'll pay significantly less than a cent for each captcha.