"Subversion uses copy-on-write when creating a branch, so it is not expensive"

Agreed - it's a constant-time operation according to the documentation, and took about 5 seconds for me to branch a copy of thousands of small files.

Git does have some useful features (e.g. committing part of a file), but a change of VCS will not magically solve all your problems if the flaw is with your workflow, code or internal communications.