So, a new regeneration. For those new to the process, this raises questions about the nature of the thing and whether the Doctor is in a privileged class among his people to be able to make use of this gift.
The spin-offs make a feast of Time Lord continuity. From the Master’s real name to that Loom business, ask any wilderness era fan about Gallifrey and instantly regret it as they regale you with the differences between a Gallifreyan and a Time Lord, the particulars of Time Lord biology and society, and the workings of their educational systems.
On TV, there’s nothing very conclusive — though there is a suggestion, which you can see in recent episodes such as “Listen” and “Hell Bent”, that there’s an extreme class system on Gallifrey, and that only the very privileged families are enrolled in the academies to become Time Lords proper.
That said, what is a Time Lord exactly? Nearly every on-screen reference suggests it’s the Doctor’s species (e.g., “The End of the World”), but that may just be shorthand for Gallifreyans in much the same way that, depending on who’s talking, “Yankee” can mean an American, a Northerner, a Northeasterner, a New Englander, someone from Vermont, or someone who eats pie with cheese on it.
The two episodes I reference above make for an interesting puzzle, as “Listen” suggests a difference between those who go into the military and those who go to school to become Time Lords — and yet as we see, Gallifreyan military figures also have regenerative capabilities.
Ignoring everything but what we actually see on the show, it may be as easy as projecting the class system to its logical conclusion. Everyone living on Gallifrey is a time lord, but only those enrolled in the Academy become Time Lords. Everyone is of the same species, and therefore has the same biology and theoretical potential, but those who go through the Academy learn some deep and fundamental lessons about themselves and their relationship with time (starting with the Untempered Schism) that give them more access to their inborn abilities — not altogether unlike attending a Tibetan monastery, as I’m sure Barry Letts would passionately have suggested. (See Planet of Spiders.)
The Doctor, being the Doctor, was not a good student. Ergo, his command over his higher biological and metaphysical functions is… lacking, compared to many of his peers. He attended Academy, yet shares more of a mind with the peasants left to scrape for subsistence in the wastes outside the Citadel.
There may be more of a reason to that than we know, as suggested in many places (e.g., “Listen”). There seems a strong suggestion that he came from a disadvantaged background, and only made it into the Academy and the associated high society through some kind of luck.
There’s no real reason to think this, aside from filling in the blanks the TV show provides, but it sounds to me like the Doctor and whatever family he might have had were somehow affiliated with the Master’s family. The Master speaks of his father’s estates, and how he and the Doctor used to run around them as children — but there’s no suggestion of wealth or privilege or family significance to the Doctor, aside from the fact of his having attended the Academy. It really gives a sense that the Master was a spoiled upper-class kid, the kind who would torture small animals, and the Doctor was the servant’s kid who, lacking any other friends, the Master took to as an assumed minion. It may well have been that association, possibly a good note from the Master’s family, that got the Doctor into the Academy to start with.
This may go some distance to explain the Doctor’s fondness for humans. He’s a time lord who may well be from an unprivileged background and, though he was “elevated” to the upper class, never fit in and still thinks of himself as one of the people.
Hartnell’s Doctor, being a recent emigrant (shall we say) when we meet him, may well still be wearing decades of this high-handed shell he was forced to adopt. It may only be after several beat-downs from Barbara that he starts to remember himself, and to allow those centuries of pretense to melt away.
For all we know from the TV show, it may be entirely possible for every person on Gallifrey to regenerate. They seem to all be time lords, with a small “t” and “l”, even if the peasants/Shabogans are barely considered people by the upper classes. They may not have the same command as the highly-trained gentry over things like regeneration and time sensitivity. It may be more hit-and-miss for them if it in fact happens at all. There may be urban — well, rural — legends of elders who regenerated hundreds of times, while others may only have one regeneration if any. Maybe the whole point of the Untempered Schism is to stabilize and regulate that business. Who knows.
But, there are lots of spaces in here to speculate. Which is, let’s be honest, a big part of the fun to a show like Doctor Who.