Git does have a server component. When git connects to an ssh remote it executes an ssh command that needs to be present.

You’re missing GitLab. I’d be looking at GitLab or Forgejo.

But you might not need this. When you access a private Git repository, you’re normally connecting over SSH and authenticating using SSH keys. By default, if you have Git installed on a server you can SSH to and you have a Git repository on that server in a location you can access, you can use that server as a Git remote. You only really want one these services if you want the CI pipelines or collaboration tools.