Somebody must own a commercial chambered 30-06 bolt rifle..you can check multiple rounds, or all of it,, of the LC blank ammo and see if it chambers,,or not.

If it does,,you've removed the idea that the ammo is out of spec.

I'd then look at the M1 rifles chambers.

I would think from the descriptions of the blank ammo used and how 'dirty' it is that the ammo is leaving a substantial layer of fouling on the walls of the chamber to build up.
Carbon deposits or what ever fouling it may be, something on the order of the carbon that can build up in the gas system/piston face when standard ammo is used.
In that case it takes a lot of rounds to build up to the point of being a thickness that needs to be scraped to be removed.
But the blanks seems to be so much more capable of layering the chamber with a deposit that would quickly change the spec(s) of the chamber in some manner so a blank cartridge wouldn't chamber properly,
It only takes a few .000 build up to do that.

If the deposits are as hard as to need a scraping to remove them, then a brushing and wipe during cleaning time isn't going to do the job in restoring the chamber to a clean state.

Boiling water treatment?,,something like the Outers Foul-Out bore/chamber cleaner?,,do they make a carbon/fouling remover reamer for the 30-06 like they make (made?) for the .22LR?

If you resize the LC Blanks in a small base die and they do chamber OK,,I think that's telling you that you have a carbon deposit build up in the M1 chamber(s) near the base.
Resizing smaller than spec is just avoiding that buildup. But as the buildup increases,,the amt the die resizes the case down will not be enough and the problem will come back.

Just my thoughts..