Others have said how "they" do it, as in production. It is also easy
to plate copper onto carbon, and then solder (or ultrasonically weld)
a lead to the copper. I've electroplated carbon rods for a carbon-arc
torch in a coke bottle using copper tube for an anode and copper
sulfate solution as juice.

I assume you've done this. Does the copper carbon bond have good
mechanical strength? most servo motor brushes have a very small area
on the end of the brush.

Karl