+1 for Brian's suggestion on cutting away the damaged area. This will help you get a clean edge to glue a replacement block in place, which will improve adhesion between the old and new surfaces. The less gaps and the tighter the fit, the more securely glue will hold it. That's probably your best bet without investing tons of time and effort.
You could also try filling in the holes with wood putty if it's a small enough area, and then try screwing in the mounting hardware somewhere else along the piece.
But the sturdiest thing is probably to replace each component as it fails.
MDF is a terrible substance to screw into because it doesn't hold mechanical fasteners well, as you've seen. It's usually better backed up by something that can handle higher stresses.
If it's not cost-prohibitive or you have some tools around, you might consider replacing each failed unit with something stronger like plywood. Otherwise you're probably going to keep running into this. The only downside is it's much more involved than a simple repair and it will probably alter the finished look unless you can manage to use the existing material as a surface layer on new construction.