"Would you have liked to have been" vs. "would you have liked to be"

It's grammatical, but it's probably not what the journalist intended. Would you like to have been president... asks whether election in the past would please you as you are now; would you have liked to be... asks whether a 'you' in the 19th century would have wanted to be president; would you have liked to have been... is asking whether an election in 1862 would have pleased you as you were sometime between 1863 and last year.

I think your observation is sound.

I know these of Conditionals:

(If you were given the chance), would you like to be President? - Present Conditional

(If you had been born in that century), would you have liked to be President? - Past Conditional