I have a problem keeping the paint on old doors that have been varnished or shellacked some 50 years ago. Even if I prime it with Kilz they still chip. Do I have to strip them completely or have you got a better idea?
Is the Kilz chipping off the surface, not adhering well? Or is the chip very thick; varnish, primer and paint?
If the chip is down to the wood; Well, after 50 years the varnish, or whatever, is very hard and brittle. There isn't anything you can about it without stripping the doors and starting over.
If the Kilz isn't adhering well then either a different primer is needed or more prep (sanding and cleaning). I would still stick with an oil based primer. But, more sanding will help any primer stick. Another major cause of primer failure is surface contaminates. After sanding wipe down the door surface with denatured alcohol, this will help clean the surface before priming.