Shoot with a gent who is quite meticulous in developing the best shooting load for a rifle. He's a retired chemist that likes to shoot on our 400-600 yd range with his rifles. About the only time I see him on our 200 yds or less range, he's zeroing a scope or developing a new load.
Anyway, whenever he's satisfied with various reloading factors for the most accurate load, the last test he performs concerns the primer to use. He'll try various brands/types of primers to see if it makes any difference in the loads accuracy/consistency. What primer generally performs best for him in his final test? Federal Gold Medal Match.
As for me, I've done similar experiments with primers. My conclusion is yep, a brand/type of primer can make a difference. Best example of difference for me was the time I used around 4 different primers and shot sets of three each of them. Shooting conditions were very good....no wind. End result on my 100 yd target was a vertical hole about 2" long and not quite 1/2" wide. I allowed my barrel to cool between each difference set of reloads. Rifle and load I was shooting in it, capable of 3 shot cloverleafs. Each primer had hit in a different vertical location on my target. Accuracy was good with each of them, they just hit target in a different spot.