But these sub-ICAO classses would need a permit for each airspace - not feasible today unless you go illegally and hope nobody notices (which is a reasonable policy).
Didn't one person circumnavigate the globe in an RV4, and another flew one to Antarctica (where he ran out of fuel, if I recall). It must be possible to get clearance, although tedious.
When Alex Henshaw did his flight he used pretty much the most up to date technology of the time. It is ironic that taking that approach now would be an exercise in monitoring the automatic systems! To make it a challenge you would have to revert to basics, but how realistic is that approach. Better to leave men like Alex Henshaw with the fame and glory that they so richly deserved. They were men of their time and we will not see their like again. But, new and different challenges await much higher and further.