I've never used Remington primers. But I'd be surprised if 1000 of them are so far out of spec that they can't be seated to a safe depth.
Keep in mind that for optimal sensitivity, primers should be seated with enough force to compress their outer shells slightly. This presses the primer compound more nearly into contact with the anvil, which makes it easier for the firing pin to ignite the primer.
The trick is to apply firm but steady, gradually increasing pressure as you seat the primer, until you feel the primer bottom in the pocket, and then just a bit more pressure to precompress the anvil and make sure the surface of the primer is below the plane of the case head. Depending on what tool you're using, it may be fairly easy or very difficult to get the feel right. Many tools that use the full leverage of a reloading press to seat the primers have poor feel.
Of course, you don't want to actually CRUSH the primers. But you do want to compress them slightly. Wear safety glasses, of course. If you're careful to make your movements smoothly, gradually, and consistently, you shouldn't have a problem. In about 55 cumulative years of reloading, neither I nor my father who taught me have ever set off a primer while seating.