[gruez]

That's true, although google's own recommendations are to add the scripts on every page

>reCAPTCHA works best when it has the most context about interactions with your site, which comes from seeing both legitimate and abusive behavior. For this reason, we recommend including reCAPTCHA verification on forms or actions as well as in the background of pages for analytics.

https://developers.google.com/recaptcha/docs/v3

A privacy minded programmer/company might restrict the scripts to the minimal set of pages, but I'd imagine most sites would blindly follow that advice and put it on every page because they think more data = better.