It seems that for some users (or at least for me), trying to POST to /oauth2/v3/authorize using the documented approaches here results in the user being forwarded back to the sign-in page with this error message:
Recaptcha is required.
This differs from the SVG-based CAPTCHA login flow that others have documented previously.
It doesn't seem like there's a workaround to this unless you use a webdriver and track redirects.
The documentation needs to be updated to reflect this if anyone else is experiencing the same behavior.