The bi-regen might be a clever way of sidestepping the timeless child nonsense.
What if at some point in the past a nondescript being (The Timeless) regenerated into two, except one went on to become The Master and one decided to become The Doctor and simultaneously lost all their memories and immortality?
This would mean that the Doctor as we know them, the madman in a box did truly start with Hartnell. It would explain how The Master is actually their brother as previously hinted. It also allows for offshoots like the Valeyard.
And yet it also keeps the Jo Martin lore intact as well, only she wasn't 'The Doctor' but a version of the Timeless-being who is no longer with us.
Sorted.