No. You are assuming that my backup and syncing needs overlap with my git usage, so that all I need to backup or sync is to use git push in my already-existing git directories.
I backup and/or sync things that I don't have under version control (git). I only sync my grocery list, for example, since I don't need a history of the 'evolution' of my grocery list. Actually, I may have it under a backup scheme, but that is only because of other files and directories in that directory that I need to backup, and I've at this point set (and partly forgot) that backup plan.
My /home/Dropbox contains a lot of files that I regularly need, and it seems a bit excessive to put all of that into one, monolithic git repo.