It took almost 4 hours, but I finally moved everything from Gogs to Gitea. The migration path was not entirely smooth due to a combination of things, but I'm reasonably confident it is actually working now. Hopefully nothing unexpected surfaces later when I've forgotten everything I've done today.
The first course of business was to regain control of the server. Every service on the machine was unresponsive, including SSH. The only thing to do was perform a remote restart and wait for things to come back online.