From every thing I have read and personal experience (M1A AR-10 Garand) it has more to do with seating depth of the primer than primer sensitivity or thickness.
Once the primer composition is fixed, primer sensitivity is set by the gap between the anvil and the primer cake under the cup, and the cup thickness. Manufacturer's don't like changing primer composition, but Federal is proud to say they use a slightly different lead styphnate, which is a little more sensitive.
I have read the nonsense about "seating the primer deeper" on one M14 forum. This is an example of the wierd ideas that come from people interpreting the world through lies they are too incompetent to understand.
Garands were extremely hard to acquire, at best you could get a NM Garand if you went to Camp Perry. If they were selling them to civilians. When civilians finally got their hands on the things, they started having in battery and out of battery slamfires. The civilian community did not know about primer sensitivity, a primer was a primer was a primer. There was no concept of primer sensitivity in the popular press of the period, and no understanding that semi automatic mechanisms would and did ignite primers due to the inertial impact of a firing pin on a primer. The only primers available on the market were the sensitive commercial primers, which were used primarily in bolt guns.And, civilians copied bolt gun reloading practices, such as neck sizing, for ammunition used in their Garands. A case that is too long, too fat, and has a sensitive primer, is creating the perfect conditions for an out of battery slamfire. But no one was warning the public at the time. The only accepted causes of slamfires were shooter negligence, that is high primers and your worn out receiver bridge. And that came directly from the National Rifle Association and the retired Army Ordnance civilians working there as technical writers.
The Army, through its minions in the NRA, put out disinformation about the reasons for slamfires in Garands. The story was, slamfires were only due to shooter negligence, that is "high primers" and "your worn out receiver bridge". It turns out high primers are the most common cause of misfires. If the anvil is not set on something firm, then the primer will misfire. You will read all sorts of threads where the reloader is having trouble with misfires on the first hammer strike. And then the round goes off on the second hammer strike. The first hammer strike is seating the primer.
This is has been twisted to a weird cult belief that if high primers are bad, then deep primers are good. So on the M14 forum, ignorant ideologues promote this idea that the deeper the bunny hole, the safer the bunny. But they really don't know what the heck they are talking about. If the firing pin can rebound off the primer, then there is the potential for primer ignition. If the firing pin cannot rebound off the primer, then the firing pin also can't touch the primer when the hammer strikers the firing pin. The end result, no primer ignition when the trigger is pulled!
There is a way to create a dangerous condition with high primers, and that is put something under the primer, such as Wayne Faatz did in the American Rifleman article "The Mysterious Slamfire". Wayne could not get his high primers to fire, when he dropped a Garand bolt on them, until he started sticking spacers under the primer. Then he had the condition of a firmly seated anvil, and a strong impact on the primer.
I do think it is a good safety operation to ream primer pockets to depth, to ensure the primer is at, or below the case head. Primer pockets do collapse the more the case is fired, and reaming to depth is a good idea on high mileage cases. A depth no greater than 0.003" is fine. Reaming the pocket too deep, to bury the bunny deeper in the bunny hole, will result in shallow impacts, and misfires.