A number of people are giving advice on how to size/partial size your cases to "fit" the chamber. And its generally good advice, IF your problem is what THEY THINK it is.
From what I'm reading, your real problem isn't what they think it is (case/chamber length issues) and every dimensional fit problem isn't "headspace".
As I see it, based on what you say (and assuming your measurements and terms are correct), your problem is due to excess chamber DIAMETER (girth, NOT length).
Why aren't factory loads failing, when fired the first time???
The answer to that one is simple. (and something well known to anyone who has reloaded the .303 British for SMLE rifles).
The chamber is "too big around" (at the head, .003" expansion of a factory load is too much). The brass stretches, ALMOST to the failure point, but not quite TO that point, on the first firing.
The brass is "overstreched" (again its diameter we're talking about not case length, or distance to the shoulder datum line (headpsance), these things do matter, and can result in similar early case failure, but at this point its the oversize diameter that, I think, is the most important factor.)
You size it back down (some, in a standard die) but then since it has been worked so much, already, the case fails when it is again fired in the oversize chamber.
Headspace (measured for the .30-06) could be spot on, but if your chamber is oversize in diameter (again the .003" head expansion) your case life will be very short. One firing. That's what you are getting, based on what you're telling us.
Again, I'll mention the .303 British and SMLE rifles. The rimmed .303 Brit case headpaces on the rim. Headspace in that rifle is the space where the rim fits. And ONLY the space where the rim fits. That one dimension can be perfectly within tolerance, and the rest of the chamber can be almost ANYTHING, and the gun will pass a headspace check. But the oversize chamber means the case has to stretch too much to have a long case life.
And, it doesn't matter, to the designers, or military users. As long as each case fired ONCE without rupture, its "good".
Your rifle MAY be the same thing, only in .30-06. Chamber length to the datum line (headspace as measured for that round) might be fine, or good enough, but its the OTHER dimensions of the chamber that are causing your issues.
Diameter issues cannot be fixed by neck sizing the brass. Length issues can be, sometimes, but too large a chamber diameter cannot be.
Get a good chamber cast done on your rifle. Carefully check the area of the case head for diameter. It may be oversize, and out of spec. It may be right on the outer edge of the spec tolerance range. If its out of spec, Remington made it wrong. If its right on the outer edge of the tolerance, despite the issues, its not "wrong".
No gun maker is under any compulsion to make their guns "reloader friendly". Most are, and that's why we reload, but some are less friendly by intent, and others by result.
A new barrel with its new chamber probably will solve your brass life problem, but be aware that a new barrel essentially make it a new rifle, and you'll have a new set of quirks to learn to deal with.
If your rifle put every shot in the same hole, every time, but cases only lasted one firing, I'd accept that, for the accuracy you get. If it shoots like a normal rifle, then I'd look into getting the barrel "fixed" or replaced.
Good Luck