I am not able to reproduce this bug locally, and Kyle told me he can't either.
Looks to me like a javascript conflict with a third-party extension, at first glance.
But I could only tell after analyzing a site with the problem, so would need access to such an installation. If still interested to use radios, then, to see the reason of the radios not working, I would either need access to the front-end of a test site, or to the admin of the existing site or (better to a clone thereof) to shortly switch and troubleshoot during low traffic hours.
Please let me know if you are interested in radio buttons and wish to give me access to troubleshoot, or if we close this support and bug report, as it works for me here, and for you without radios too ? If interested, then please Private-Message me (link below) access information (URL, username, password). Thanks!