I am pretty sure you're missing something here, but don't have time at this moment to sort it out. However, I will point you to Flightwise, Volume 1, page 210:...
I wouldn't necessarily disagree with the flavour of Carpenter's conclusion. After all, if skin friction Cf varies as Re^(-1/5), that simply means that skin friction drag actually varies as v^1.8 rather than v^2. Using a constant Cf is probably good enough for government (or Airbus, or Boeing
) work.
Anderson derives the skin friction drag coefficient for laminar flow and quotes the result for turbulent flow in Chapter 17 of 'Fundamentals of Aerodynamics', so all I can do is point you to that if you're not convinced.
Abbott and von Doenhoff also derive the turbulent case in chapter 4 of 'Theory of Wing Sections'. They reconcile the results with experiment which demonstrate a slightly (but not substantially) weaker dependence on Re.