I am a mechanic, with a degree in mechanical engineering. I actually invented the ICE. I am also an attorney (my experience includes writing our constitution - my "john hancock" is right there at the bottom. Therefor, my expertise is beyond question, so none of your opinions matter; you must all accept my opinion as gospel or you are mentally ill.
[sarcasm off]
It is my professional, expert, knowitall opinion that the number one value of a carfax is in the asking - if they don't promptly show it, that's enough of a red flag for me to go look elsewhere. The only exception I can think of is the private seller; I can see why someone wouldn't want to pay for it in such a circumstance. I don't agree, but it's not as big a red flag as a dealership or other car sales lot not having it available.
Because my Expy was a government truck before I got it, the multipage carfax that came with it did show a lot, but, as others have mentioned, it did not show everything. It did show, however, that the spark plugs had been changed, so that was a benefit right there. On the other hand, it showed some major brake work well ahead of expected service life. I spent considerable time trying to get to the bottom of that, without result. Oh, well.
Like so many other things in life, there's no categorical answer as to whether or not a carfax or such is an absolute value add. It's your car, you get to decide.
As for me, not only did I get a carfax, I also took it to my mechanic, after my own inspection. Paid off handsomely.
You heard it here first.