After choosing your Origin, a new selection layer will appear asking you if you want your Origin to be randomized after each death.
Custom Origins are supported as long as they are allowed to be randomly chose in your Origins layers settings.
There's also an alternate version of this datapack that supports the Origins: Classes mod (as a result of the way datapack works, you can choose a random class on the class selection screen even without activating the randomizer).
This datapack was made in version 1.20.1, but should work on most versions of the Origins mod.
Project members
The_Feness
Developer