Primer in the countersinks or dimples won't affect anything, unless you apply way too heavily. Most people prime too heavily; it's not a top coat. And, unless the primer you use acts as a moisture barrier (not all do) then a heavy coat really buys you nothing. If you get a chance, check out a quickbuild kit and see the priming job on that. Playing Devil's advocate

, if you really want to overkill it, mix up a small amount of primer and put your rivets in that and set them up wet. No, I don't really advocate it but it's a technique that has been used - with bolts as well.
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Patrick Kelley - Flagstaff, AZ
RV-6A N156PK - Flying too much to paint
RV-10 14MX(reserved) - Fuselage on gear
http://www.mykitlog.com/flion/
EAA Technical Counselor #5357