Well, through a ton of trial and error, I got it to work again.
The thing for DietPi is that the Gitea user needs shell access. I had to change /usr/sbin/nologin to /bin/bash in /etc/passwd. I know its far from ideal from a security standpoint, but that is what needs to be changed for it to work out of the box.
I also had to change the SSH command in the git repository to include the key file: git config core.sshCommand 'ssh -i ./.ssh/private_key_file'
I am not sure why it is not respecting the git configuration file, but for now it is working.