I lived through Nixon's presidency and I never noticed that particular verbal habit. He had many. (One that grated on me was "Let me make one thing perfectly clear...")
One problem with "in all candor" is that it is somewhat rare and a little pompous. It sounds like someone insecure searching for elevated language. That may be what "sounding fake" means. Elevated language may come naturally to some people--academics, for example. Or, it may be someone trying to sound important--like an administrator saying "it has come to our attention that" instead of "I've heard."
It seems amazing that a President of the United States could feel insecure or struggle to sound important, but that is EXACTLY the impression Nixon made on some people.
Another problem is that Nixon uses it to mean "Now I am about to tell the truth and let you in on a little secret." When people begin by saying they are going to be honest, it often means they are about to be dishonest. After all, if you say "frankly" or "honestly" or "truthfully" or "In all candor," you are saying that you are about to do something DIFFERENT from what you do the rest of the time--that most of the time you are not frank, honest, or candid!