What do you use for storyline file Version control tools? Git?

It's been some time since I've used git, but I think your boss is correct; it doesn't seem to be the best tool to use.

It sounds to me that your group's work processes are really part of the problem. If everyone is "needing to edit a storyline file" at any given time, it's no wonder that yoou are seeing versioning issues.

Have you tried to define specific owners for respective files/stories? At the very least, that would seem to be a good step to start with. Then you might try creating an unpublished scene in each story that contains a simple list/table:

Date     Author/Writer/WorkerBee         Actions Taken        Version Saved As

And while the honor system may presently not be working, this sort of tracking would at least allow you to have a pretty good idea of who toggled a variable in scene X that caused scene Y to stall...