With Git, for example, you modify the code on your machine, then save it, that you have to commit and push. Or am I wrong?
Your development machine should be able to run the same code as your server.
Making changes to the public site is a very bad practice. What if your code has errors in it?
Version control also keeps everyone up to date. Before making a change to the site you'll download the most up to date version from the repository, that way your local copy has everyones changes, with comments about what they've done.
my standard process is so:
have my checked out code mounted via nfs etc on the dev server.
make changes.
install.
if it works correctly, check it in.
we have a very flexible build system that allows for anything from the entire system to being installed down to just a single file.