If you change the aperture/F# by a factor of 2 (2x, 1/2) it changes the DOF by a factor of 2 as well. But if you change the FL or the subject/focus distance by a factor of 2 it changes the DOF by a factor of 4... they have 2x the effect that aperture does, only in opposite directions from each other.
"So what (correct) statement can be made to compare ...
What your friend told you is a way to estimate background blur, not depth of field. If you look at the DOF formula, you'll see it's inversely proportional to f2/N, not f/N. This corresponds with your statement, "focal length has an higher impact on DOF than the F-stop".
DOF = 2 u2 N C / f2
N = aperture F-number
C = circle of confusion
u = ...