The whole point of using git is that you don't have to do this kind of crazy stuff.
I know what you've said about why you don't think you should all have complete copies. Here's the hard truth. You're wrong. Mostly. But that's ok, you said your willing to rethink how you work and that's good. Ill try to explain why its not that big a deal to have everyone use their own clones.
A assume all code is already in a remote repository - if its not, sign up on http://github.com or http://bitbucket.com and get a free repository, add it as a remote to your git repo, and push it up. Its really very simple.
Each of your developers should then make their own directory locally on their machines where they can clone the whole repository.
git clone http://github.com/yuoraccount/yourrepo ~/clones/localproject
The first time they clone, it will take a little time to download everything, but from then on, only each minor diff will need to be downloaded uploaded. Git is made to be efficient that way.
When you make a change, commit it, and push it up.
git commit -am "i made a small change"
git push origin master
Then everyone can pull it down.
git pull origin master
You can even all work on different branches, so your not just pushing to the same branch. This should all be really simple, and very easy to do.
You can also split your project into multiple repositories, but you don't really need to. Thousands of files is not a big deal, git can handle it without a problem. That's not to say that you won't have some challenges. Git is easy to use, but you'll eventually run into merge conflicts. They will be a little bit frustrating at first, but stack overflow has a thousand answers explaining how to deal with them - you should be fine.
This is what git is for.