Sorry, haven't read this entire thread but can offer some use experience on White River primers. Here in NZ we had the same primer shortage nightmare and I had to resort to any alternative when my preferred Federals were unavailable. So far I have used only WR LR primers in mostly Norma brass. Yes, WR cups are deeper than other brands ( well documented by others online experience / comments ) and I need to use a little more muscle, or two priming pushes, in my RCBS Benchtop Priming Tool to seat under case head level. Only a couple wouldn't seat below head level and needed putting aside. So this is with Norma brass. When I experimentally tried some Remington and Sako brass the primers easily seated below head level so that may be a solution for some handloaders.
I can say shooting White Rivers has been great. I'vs now used several hundred and continue to use with confidence. I even did some comparisons against Federal 210 primers in 243W and 30.06 loads ( same brass batch / prep, powder, projectile, COAL, shot through same rifle ) and chronographed velocities and POI are near enough to identical. Never a dud or misfire. I think they are very good primers.
I also bought a few hundred WR Magnum primers but haven't tried any at this stage.