I see no reason to use a super hard military primer. They would work just fine, but nobody I have ever heard about has had any trouble with CCI, Win, Rem, or any other LP primer I can recall. The only primer I would not personally use, whether for factual reasons or not, are the Federals. I know from decades of using them that they use softer primer cups than all the others and honestly, they scare the hell out of me in a semi with a floating firing pin. I only use Federals in bolt, single shots, and revolvers and there they have no peer IMO. I love them. However, they show signs of flattening at start loads that the unfamiliar could interpret as being high pressure signs, so you cannot read them like you do all the others. That's why I know they are soft so they will not be used in any semi with a floating firing pin, not by me anyway.
Since I have always be under the impression that in terms of 'hotness' regarding primers, magnum primers are hotter than standard and rifle primers are hotter than pistol. If you reamed out the primer pockets to take LR primers, what do you gain? Nothing. You still cannot exceed 36.5-37K psi or you could suffer bolt lug failure from excessive back thrust. And by using pistol primers, if you see them flattening you know you are already well over 37K psi, likely around 44K psi, but with a rifle primer you could be pushing 65K or more and your rifle could turn loose on you without the primers ever indicating that things were getting a little hot. Little hot hell, a whole lot hot!
Another thing I have always read about is that the use of magnum primers is like adding another grain of powder. I'm not sure I'm buying this because I have substituted standard primers in place of magnum primers and saw no change in velocity (going from magnum to standard, I have never tested going from standard to magnum). Maybe if it were really cold I would have but still, "they" say reduce the maximum load by 1 grain if it was worked up with standard primers and you are loading magnum. Better safe than sorry so I do it.
I have also read that switching to rifle primers from pistol primers does the same thing. I have never tested that but my own paranoia would say if the maximum load had been worked up with LPM primers and I was switching to LR or LRM primers, especially LRM primers (after reaming the primer pockets of course) I would either drop the maximum listed charge by a grain or two or use the chronograph and stop when the velocities were the same between the load using LPM and LR/LRM primers. I would not try to exceed the velocity you were getting with the LPM primers.
I have always heard that CCI primers use the hardest cups with Winchester being a close second. In ARs and LRs, I always use the standard CCI primers, by standard I mean non military. In my 6.8s and their wildcats I use SRM, the 5.56 uses SRM, SOCOM based rounds all get the LPM, .338 Spectre (10mm magnum based) gets LPM just like the SOCOM rounds, and my LR-308 sized wildcats and .338 Federal all use LRM primers. Sure, I get primer pecks when you drop the bolt and chamber a round, but I see that even on military milsurp ammo and the peck does not look any more shallow to me with the military primer than with the CCI primer. It certainly is not enough of a peck to give me any concern and I am a pretty paranoid person.
In a bolt action where you can safely push pressures to 50K psi or more, sure, ream away and use LR or LRM primers, but in an AR I personally would not do it. It may lead you into a false since of security with your loads and that could get you into big trouble very quickly. Using the LR primer is not going to allow you to shoot hotter loads and IMO they really do not reduce the risk of slam fires. I've dropped the bolts on thousands and thousands of rounds of SOCOM ammo and never had I had a problem with a slam fire or anything close to it using pistol primers. And if you practice safe gun handling, having a slam fire is not going to do anything other than scare you shitless. Always assume when you chamber a round in anything that it is going to go off. Do that and then if and when it ever did happen, no big deal. Under ware can be washed.
The only thing I have found in regards to the use of pistol primers, I think every one of my ARs in a SOCOM based caliber has had an issue with intermittent primer piercing, sometimes even on mild loads. In every case the firing pin was within mil-spec, .028-.036" protrusion, but IMO a firing pin over .030 is too long for the softer and thinner pistol primers, and that causes the piercing. I use the standard firing pin until it pierces a primer, then I buff it out to remove the erosion and set the protrusion to no more than .030 inch. After that, no more pierced primers, ever. I have never had what I would consider a safety issue with pierced primers other than the pierce, no blow back, no bolt face erosion, nothing, just after shooting I'll notice a burn through in the primer dimple and of course, the firing pin tip will be eroded if you look at it under magnification, even with just one piercing. Rifle primers might prevent that with .032-.036 protrusion firing pins but for the reason I mentioned above I would never use them in the AR.