Couple of things...
(IMO)......
I clean my pockets every reload and check the flash hole for obstructions... I don't uniform unless I reload brass that I buy (or collect at the range from others......
I'm a 'brass farmer'....). Then I uniform them the first time and never after that.
If the pocket is the correct diameter and it retains the primer properly, in my view, the firing pin isn't going to move the primer if indeed, the pocket is too deep because the firing pin is an impulse hit (like a karate chop), not a push. If the flash hole is open with no impingements, the primer flash off will travel to the charge, even if the primer isn't fully seated against the base of the pocket...
The mechanics of a primer is it's actuated by a sharp strike in a concentrated area, so....
Typical primer indent should be .004-.006. It's hard to measure so I'd go with factory stickout specs on the pin as it relates to the bolt face.....
Of course there is the headspace issue and if the cartridge is seated fully against the bolt, but if it extracts properly, chances are it's sitting correctly against the bolt face in the first place. With my semi-auto's, excessive headspace or COAL will cause erratic cycling in a bolt rifle, the bolt won't close properly or you'll feel resistance when you close it....
I like to hand prime versus machine prime because I can 'feel' the primer being inserted into the pocket and I can 'feel' with my finger if it's seated to the correct depth .002-.004" below the base of the cartridge. You develop the 'fee' as you load. I'm a sit in the living room and prime person.l
If the primer seats with resistance, it probably won't move upon impact, I say probably because anything can happen, this is real life, not a textbook...
I'd suspect firing pin stickout or excessive lubricant on the pin as it rides in the bolt.. I'd be checking stickout with a caliper and depth attachment to insure it's within factory specification and throughly clean the bolt and mechanism, I tend to use brake cleaner myself. It's cheap and removes all the oil. The pin needs just a tiny amount of lubricant, a small drop. Thats it.... and a non coagulating lubricant. Some lubricants get more viscous with age and heat. Read the bottle....
We all like to keep our firearms oiled and looking good, however, the firing mechanism needs lubricant applied sparingly, especially the firing pin and the bore or slot it moves in.....
In as much as I don't know the manufacture of the rifle I have no idea if the actuation spring is light or heavy tension but in the case of Savage, the factory uses a light gage spring so excess lubricant can slow the inertia of the pin strike and cause mis fires.