There is nothing at all to be paranoid about as regards the Lee-Enfield rifle and headspace.
The REAL issue is a general lack of understanding regarding what "headspace" actually IS.
The American gun magazines ascribe sloppy headspace to anything not "Made in USA" or Germany. British, Japanese, Italian, Russian: NONE of them are up to snuff by their standards. The biggest problem is not the rifles, but the American gun magazines, which my long-time shooting partner (one of the finest handloaders and riflemen I have ever known) referred to as "the funny papers".
HEADSPACE IS THE DISTANCE BETWEEN A CHAMBERED ROUND, AS FAR FORWARD AS IT CAN GO, AND THE FACE OF THE BOLT IN LOCKED POSITION.
RIMMED cartridges headspace on the Rim of the chambered cartridge; extra space in the front of the chamber can be a nuisance but also is a great feature under battlefield conditions. It is NOT a weakness and it is NOT dangerous.
SEMI-RIMMED cartridges headspace on the small Rim of the casing which, as with Rimmed ammunition, prevents the cartridge from going too far into the chamber. Again, free space in front of the round is not an "issue" although it can be either a nuisance or a lifesaver, depending on your point of view..... and if you are in a mud-filled shell crater with somebody nasty shooting at you.
RIMLESS ammunition headspaces on some theoretical slope of the shoulder of the casing and depends on careful chambering and precise fitting of the Bolt to the barreled Action. There must be ZERO room, ideally, in front of the chambered round and the Bolt must close solidly against the cartridge. Because this was nearly impossible to achieve before CNC machine-tools were developed in the 1950s, the Americans allowed .003" of extra space and looked mightily askance at anything more than that.
Considering these points, I have NO idea why anyone would want to make a rifle to handle Rimless ammunition at all! It is just too much trouble! CLEARLY, the best compromise as far as rifle effectiveness is concerned, must be the Semi-rimmed cartridge...... which was used by the Japanese Arisaka and by the British P-13 which was developed as a wartime expedient into your P-14 and chambered for the regular rimmed .303 cartridge.
Headspace with the .303" cartridge is simple: .064" is Minimum, .074" is Maximum. Cartridges were made with a maximum Rim thickness of .063", so a well-fitted rifle would have exactly ONE thousandth of an inch of "headspace". If this were more than TEN thousandths, the bolt-head (on a Lee-Enfield) or the Bolt (on the P-14) would be changed-out to a better fit.
You CAN experience "headspace issues" with some modern ammunition, even with a perfectly-fitted rifle. Manufacturers do not always stick with the miltary-spec RIM thickness (.060" Minimum, .063" Maximum), either out of sloppiness or misunderstanding of the criticality of rim thickness with Rimmed ammunition. I have seen commercial ammunition on the Canadian market (many years ago, mind you) which had Rims as thin as .040", which meant that every single round would give .023" of headspace OVER and ABOVE what the Rifle gauged. A rifle with .010" of actual headspace would be running .033" when using this ammunition: about FIVE TIMES what the Americans consider dangerous...... and there were no serious accidents, although much of the brass wasn't worth picking up.
As to the P-14 rifle, it is one of the finest rifles ever built. That massive action will handle literally any cartridge which can be fed into it. They are immensely solid, smooth and lock up like a bank vault. The ONLY weak part on the rifle is the little spring which powers the Ejector; if it breaks, it can be replaced with half of a spring from a ballpoint pen and it will run the next century without a bobble. It has a pure Mauser action and feeds FROM THE MAGAZINE, so don't toss single rounds into the chamber and try to slap the bolt shut.... unless you really like buying Extractors..... which are very nearly non-existant. Once the Bolt grabs the cartridge, chambering is quick and slick and most ammunition problems disappear, even with that horrid rimless .30-06 used in the P-17.
But if it happens to have a chamber which is a bit deep, that is no "issue", nor is it dangerous unless your ammunition is the worst kind of garbage. It is simply the way the rifle was designed and built. To compensate for a deep chamber, as with a deep chamber on a Lee-Enfield, all you do is neck-size your cases once they are fireformed to your rifle.
NO problem.
GOOD rifles..... all of them.
.