Now that the dust has settled, and I've had a bit more time to think about the episode, there's one aspect of The Doctor's past that I haven't seen anyone yet comment on:
Where does Susan fit in all of this?
Seems everyone has forgotten that she was with The Doctor when he fled Gallifrey. Why did he take her? Who was she?
So let's see what I can come up with for "The Timeless Children" that preserves what we knew before the episode, most of what we saw in it, while getting rid of the worst stuff.
We know that:
The Doctor was involved way back in the beginning of the Time Lords and Gallifrey. His exact role is unknown, but he ranks right up there with Rassilon, Omega, and "The Other" in importance.
He said the reason he left Gallifrey was that he was exiled - but later on it seemed as if the exile was more a matter of his own choice than his being forced to leave.
He might have even served as president for a while; whatever the case may have been, he certainly held a very high office in their ranks.
Under normal circumstances, there is a limit to the number of times a Time Lord can regenerate - and they can regenerate at will, if they want.
Now, if (what I recall of the events in) Lungbarrow is canonical:
Way back in the beginning, there was some sort of major crisis that led to a huge decline in fertility of the Gallifreyans. Instead of the normal process, they came up with "looms" to essentially artificially reproduce. The Sisterhood of Karn had something to do with it, as well as the whole process of regeneration. Susan was somehow a key figure in the events.
So.....
Way back in the beginning during the Fertility Crisis (but after time travel had been discovered - and Rassilon laid down the law about looking in to the future of the Time Lords and all that), The Doctor was working with The Division, an elite government research team, on coming up with the solution. He invented / discovered "regeneration" (and was able to give it to himself before the gov't / Time Lords found out - and put limits on the number of regenerations). When he found out that The Division was testing the process on children (maybe he was given access to that information when he became president?), he grabbed one of the test subjects (Susan), stole a TARDIS, and fled. He would later give the Sisterhood of Karn the secret of unlimited regeneration - just in case. Presumably he gave the ability to Susan as well.
When the Time Lords found out about his flight, they slapped a sentence of exile on him ("If you ever set foot on Gallifrey again....."). Later on, however, after a change in the government of Gallifrey and the Time Lords, the sentence was lifted. But for personal political reasons, The Doctor prefers to have as little to do with them as possible.