Visit every country in 72 days? A UN senior official might be able to manage it with one of these: https://en.wikipedia.org/wiki/United_Nations_laissez-passer , which would also provide a way round the mutually-exclusive visa rules. You might encounter some problems with defining a country (Palestine, Taiwan, Nagorno-Karabakh etc)

Physically it would be doable with a private jet and some helicopter transfers for the awkward cases. Microstates without airports (San Marino) and tiny islands (Sao Tome etc) might take up a lot of time.

(The Verne trip was a "mere" circumnavigation, immigration was a lot easier back then, and quite a lot of it was British Empire)