I had exactly this problem for the last two weeks and tried the workarounds but with no luck. But I then wondered if the Brave shields/JavaScript enablement themselves were stopping the captcha from appearing.
Basically I disabled all the shields and relaunched Brave, clicked ‘Solve’, the captcha appeared and I was finally able to complete it.
e.g.
• Go into 'Settings' | 'Shields'
○ Change 'Trackers & ads blocking' to 'Disabled'
○ Change 'Upgrade connections to HTTPS' slider to 'Off' position
○ Change 'Block scripts' slider to 'Off' position
○ Change 'Block fingerprinting' to 'Disabled'
○ Change 'Block cookies' to 'Disabled'
• Close all instances of Brave, then relaunch it.
○ The Captcha window should open again, but when you click 'Solve' this time, it displays the captcha and allows you to solve it.
• Set all your shields back to however you want.