CF7 only works when using version 4.7

Are you using the reCAPTCHA Integration setup at Contact Forms > Integration on your site?

Yes, I am using that. After I upgraded to the latest version of CF7, I created the new V3 keys in the Recaptcha admin console. But that didn’t work either. So, I switched back to version 4.7 without Recaptcha for the time being. I am struggling to find out what causes this behaviour, as usually, I have no problems with CF7 whatsoever.

CF7 5.1.+ updated to reCAPTCHA (v3).

If you registered correctly for reCAPTCHA v3 you should see the reCAPTCHA Privacy badge displayed at the bottom of every page on your site (not just the pages where you’ve got CF7 forms).

See List of answers to issues with adding reCAPTCHA v3 to Contact Form 7 which covers this topic in detail (disclosure I’m the author of this article).