Works for me. Assuming one uses something like Privacy Badger and it if it were programmed to block reCaptcha, these websites that require recaptcha will go the way of anti-adblocker popups. People will simply say no and hit the X and go to their competitors.

Sure, my gov (Brazil) uses reCaptcha on the page where you can check your electoral status (For example: if you can vote, where, and if not, what is missing). Where can I find a competitor for that?

Ask your political representative why they're relying on a foreign ad service to manage their government websites.