Excess headspace does not cause primers to fall out.
It only causes short case life for reloaders because the cases will break in two from excess stretching. On high-pressure rounds like the .270, the case expands and stretches to fit all the excess headspace, and stays that way with the primer still seated correctly.
Primers falling out can only be caused by excess pressure expanding the casehead & primer pocket beyond it's ability to recover. Or reloading the same case over & over so many times the primer pocket just wears out and gets loose.
The first thing I would do is inspect the rifles bore for excessive copper fouling.
See if there is still rifling visible, or if it is completely filled with copper fouling from bullet jackets.
The next thing I would do, after a very through cleaning with copper solvent, is slug the bore.
Driving a lead slug through the bore and then measuring it with a micrometer will tell you if the bore is actually .277" or not.
It could be Parker-Hale screwed on a .25-06 barrel and chambered it for .270 Winchester.
The third thing I would do, if the bore is in fact .277" is make a Cerro-Safe chamber cast and see if the throat or leade is cut properly. A too short leade will cause the bullet to jamb into the rifling before firing and pressure will be much higher then normal.
You can also check it at home by coloring a bullet with a black Magic Marker and then chambering it.
When you eject the round, look at the bullet and see if the rifling scraped off the black ink.
If it did, you have a too short chamber throat or leade.
Should that prove to be the case, a gunsmith can run a .270 chamber reamer in the chamber and cut the leade properly.
rc