For people who make a living as an impersonator, like Frank Caliendo and Rich Little, capturing the unique identifying characteristics, tics and vocal peculiarities of well-known personalities and casting them in a humorous light is much more important than creating a completely believable character. They are comedians, not actors. Believable isn't the point for them. Recognisable and accurate is.
Anytime you play a real, historical person on stage, particularly people we've seen and heard on video or film, you risk becoming an impersonator rather than an actor. It is easy to be so concerned about being faithful to their external nature that you forget to do the extra work required to find the inner person who manifests those externals.
Playing real people on stage is very challenging, just as talking directly to the audience is. Combine those two things in a one-person show, and you've got your work cut-out for you. Line reading becomes a very strong temptation is those situations. After all it's unnatural to speak to people who never talk back, who don't respond "in character", because they aren't character, they're audience.
There's an on-going debate among actors as to it's better to start with internals and move to externals, or vice-versa. It doesn't really matter where you start as long as you approach the internal aspects in an "organic" way. Whatever triggers that for you is fine, if it works.