Ok. So check it out.
My buddy was playing a new character, and I asked for a boost on my new Moze. So I joined him in competition mode. He was fighting Troy, and after Troy you get the eridum analyzer. When I went back to my game, I had the analyzer WAAAY sooner then I should have. (While also keeping my fast travels from him.)
So. In theory. If I join a TVHM player, with my NEW TVHM character, as they complete the story for the first time, I may unlock mayhem mode early?
We plan to test this theory when he gets to Tyreen, I will join him with my level 10, and if it works on normal it should work on TVHM.
The theory is, you could potentially unlock everything using a friends game, then just need to complete story missions on yours.