No, Google Site Kit was already setup when I made the staging site. I also did not migrate the staging site to the live site. I simply deleted the staging site, and then GSK broke on the live site which is why this is confusing. Since they are supposed to be totally separate, I wanted to understand why this would have happened. And when I started to go through resetting it, it sent me to the staging URL instead.
Yes, the plugin is out of date. I recently replaced another developer so I have been updating plugins, and that was on the schedule for yesterday – but couldn’t since the GSK broke on the live site.