Thanks for your replies, guys.
Youngeyes,
I agree, the punch should extend into the pocket. It does damage the case by scraping/shaving the inside of the primer pocket, as if the punch needs to be machined or polished down a little.
You are also correct that I should not have so many missed primers, and I have corrected the original problem which was causing this (I had loosened the lock screw for the punch/cup assembly, but hadn't reseated them properly). After figuring that out, I checked it to see if it would still damage a case, and it does.
Yes, I am new to this platform. I do have a fair amount of experience with reloading with single-stage presses. All of my priming has been done with the Lee Auto Prime, so I'm comparing the priming experience of the press to the Auto Prime experience. My brother keeps saying "why don't you just do all the priming on the Auto Prime?" That's always an option, but I didn't buy a progressive press just to have to do everything singly.
Shellplate is indeed tight, and I am using the correct primer assembly/parts.
Moto,
Yes, I am using some brass that has crimped pockets, and some without. Multiple headstamps: LC, WCC, Winchester, RP. I've only loaded .223 so far, as I'm waiting on a set of 9mm dies to arrive this week. So all small primers (CCI #41s). Case prep consists of tumbling, One-Shot lube, full-length resize/decap, trim to 1.75, chamfer/deburr case mouth, then use the deburring tool to remove crimps/swages (using a light cut). So yes, crimped pockets could definitely be a contributing factor, but I figure if the Auto Prime can handle it, the 550 should be able to handle it as well.
In my mind, and my brother's mind (also an experience single-stage reloader, and a pretty sharp guy), something isn't right about the punch damaging the cases, and crimped pockets don't really seem like they are the reason for this occurrence. It could account for some things, but not all.
I, too, get the primer started into the pocket carefully before I apply full force. Usually, I have to hold the back of the press while pushing on the handle, and even then primers aren't always fully seated. The benchtop I'm using consists of a sheet of plywood sandwiched between a sheet of 1/2" or 5/8" melamine-coated particle board and a 2x6.