Sorry, they can't do that. Then an attacker could do one or a small number of CAPTCHAs, and then use the now-proven-"good" account to do some amount of mischief before getting shut down, where the amount of mischief is proportional to how much Google et al accommodate your otherwise-reasonable request.
trust can be more than a boolean, and P(captcha) can be proportional to my trust. I bet of the people who have done hundreds of captchas on stackoverflow, exactly zero of them are going to bot their accounts.
