I've run into state and local tax agencies, utility companies, and large healthcare companies that require Google's reCAPTCHA. So, unless you don't want healthcare, to have water service at your home, or you're in the mood to just shut down your business, you have to suck it up.
‘Need’ here means exhausted all other opportunities, and have built alternative accessible ways of accessing the same service. I’d certainly have expected a service to have investigated a self-hosted solution, and I doubt a reliance on 3rd party JS from a Google service would fly, regardless of the service, as it breaks a whole bunch of separate resilience guidelines.
there was a time not long ago before wheelchair ramps or accessible doors were commonplace. these people were literally shut out of society.
its the same with captcha forcing privacy-conscious users off the internet.
Or: people who need a wheelchair are protected by anti-discriminatory laws, while people who prefer not to use Google products aren't.