I have asked the best shooters in the area, some are National Champs, it is very hard to sort out primer induced errors, if they exist.
There are different combinations in which you will get better accuracy with one primer or primer type. You really have to experiment.
One shooter who I really respect said he got better accuracy with small rifle primers, but the ignition was not as certain and he had misfires in cold weather.
I do notice a lot of good shooters using CCI benchrest. These are very consistent primers.
I did a primer test with Tula primers in the 30-06 and they did very well, I would say better than any, but my sample size is only ten shot groups, so that is not statistically significant.
The largest errors are with bad bullets. Powder probably next, then cases. Primers are probably in the 5% or less level.
As one excellent shooter said to me on this question: "Primers, you have to be real sensitive to see a difference with primers".
I am not that sensitive.
If I'm not shooting, I'm reloading.