You can countersink for rivets in metal a bit if you want to, but wood is too soft. You won't be able to pein enough to fill a countersink in wood.
Much better to use a tapered reamer, very lightly. This gently flairs the holes without having to try to pein too much.
This is how you set pocketknife rivets so they never come loose as well.
Just lightly taper the holes.
oh, and basic peining of rivets?
You can use the flat or the ball or both. The trick is to strike near the outside, (either with the ball or with the edge of the flat end) alternating around the perimeter of the rivet. Kind of the same as torqueing a head. Except you can make several light taps before going to the opposite side. Keep tapping, sharing the wealth, until the whole perimeter is done.
(Try to strike just inside the perimeter of course, or you will mar the host material.)
First do one side, and then the other. And then back and forth.
This spreads the rivet metal at the ends without bending the rivet.
yours Scott