We loaded our cases between each group at a bench somewhere behind the line. Some loaded in modded trailer, some in motor homes, some in whatever building or shed the range had, some outside at a table we brought with us, but never at the shooting bench. I don't know anyone who ever weighed primers. Primer weight is a non issue. Primer consistency is important, and why the best primers were sought out. The "match" primers loaded by CCI, Federal etc are done by the most experienced workers vs the standard primers. In other words, they are good at putting the same exact amount of primer compound in each cup and setting the anvil consistently. The cases were indeed highly prepped previously. We used "custom" "Benchrest" bullets made by various people where all the bullets came from the same die. Watson, Gentner, etc, etc. Lot's more makers these days. We dropped powder with careful technique but didn't weigh it. Some never weighed it ever, and no one weighed it at the match, but just dialed the measure to the right setting and charged cases.
Those rounds never had to travel more than from the bench to the line and maybe back to the bench. For rounds that are going to travel, I had a couple of ways, both mentioned in the thread. Bullet up, or bullet down where the round rested on the shoulder.