I like using Git software to push commits, but the ones I use (Gitbox, Github, SourceTree) all ask for a local repo when adding a new repo to them.
Thing is, my repo is
I had this exact problem about a year ago - unfortunately I couldn't find any consistent, reliable answer. I googled for weeks, thinking it was my search terms that weren't successful - trying every which way.
[The set up we had was each developer had their own dev server - having them separate from machines meant that sites could be developed anywhere, dev servers could be set up to be exactly the same as live environments and the sysadmin could keep them upgraded and backed up - i fully see the advantage of having a separate development server to the working machine, with one of the few downsides being no git apps!]
Everything that relies on file mounts or spoofing your computer into thinking a remote drive is local is fine for general file browsing, but Git apps tend to flake out when the connection is intermittent. Other times you have to do things in a certain way, just to see a git status
I know this is not the answer you want to hear as I was in your situation a while ago and know exactly how you feel, but the best thing you can do is use git on the command line.
I hate to be one of those command line is better Stack Overflow answerers, but in this situation I couldn't find anything that was up to scratch, to be used all day every day by multiple developers.
I was also against it at the time, I prefer the prettier, easier to use UI but since learning the command line & git, I have never looked back. When starting my own projects at home I find myself using terminal over any apps as I find many of them confusing!
It not only helps with your command line confidence, but my Git knowledge has improved tenfold since using the terminal, as the apps often hide a lot of the happenings.