Update:
If you want your repo be accessed from anywhere by anyone. You could choose to use Public Project in Azure DevOps Service.
An Azure DevOps Services public project provides support to share code
with others and to support continuous integration/continuous
deployment (CI/CD) of open source software. Users aren't required
to sign in to gain read-only access to many of the services.
More details about public project you could refer official doc here: What is a public project?
According to your description. Seems you do not want to store your "private" code files on your company's TFS server but rather on your local hard drive.
Then you should not use "Add solution to Source Control" button, this is used to connect your local solution to a server side source control.
Since you just want to maintain your local codes in git repo. You could open a project that’s controlled by Git (or just git init an existing project), and select View > Team Explorer from the menu. You’ll see the "Connect" view, which looks a bit like this:
Visual Studio remembers all of the projects you’ve opened that are
Git-controlled, and they’re available in the list at the bottom. If
you don’t see the one you want there, click the "Add" link and type in
the path to the working directory. Double clicking on one of the local
Git repositories leads you to the Home view, which looks like The
"Home" view for a Git repository in Visual Studio.. This is a hub for
performing Git actions; when you’re writing code, you’ll probably
spend most of your time in the "Changes" view, but when it comes time
to pull down changes made by your teammates, you’ll use the "Unsynced
Commits" and "Branches" views.
More details please refer Git in Visual Studio.
Another choice is hosting your local code on a private hosted Git provider. This make things more flexible. For example, it's easily to let you continue work on your private test projects, if you want to use your personal computer.
If you don't already have a private hosted Git provider, Azure DevOps Services is a great free option. It's the cloud version of your company's on-premise TFS server.
In addition to free GIT repos, it also provide high performance build/release pipeline, configure Kanban boards, automated and could based testing.
This will be a better option if you already familiar with workflow in your company's on-premise TFS server.
