I'm a geek. I think about things like this. Especially as I watch the first season again. No, I don't mean Eccleston's Doctor, I mean Hartnell's.
We know he came from Gallifrey, that isn't what I'm talking about. No, I'm talking about the inconsistencies that have arisen from where the show started to what it has become. But the genius of that first season is that there are lots of touches that when we consider what our favorite Timelord has grown into, offer some interesting possibilities for his past.
The Doctor did not originally have two hearts. That came later, after we learned he was a Timelord. Nothing about the Doctor's physiology is seen as particularly surprising until after that. He is even examined on at least one occasion by some high tech medical equipment (The Wheel In Space). But there are hints with the first Doctor, especially in light of more recent comments by the Doctor, as to what the issue might be.
We have to remember the First Doctor is old. He was seen as a very old alien at first. When you do some subtraction on the figures Romana gives for the Doctor's age, he was 234 when he stole/borrowed the Tardis. That is still a respectable age, though far less from the 600 or 400 he was originally written to be. And William Hartnell, although only 56 when he took the role, was suffering from arteriosclerosis, making him seem far older. Plus he was playing older. What we get is that the Doctor is a really old man, weak, in need of frequent rest. He does not spend much time running, he leave the strenuous activities to his younger companions. We don't get a leaping, active Doctor until after his first regeneration.
This all comes back to his hearts. He's a Timelord, we know that now. He has two hearts. Yet this man in his prime (let's face it, he is young for a Timelord) is very delicate. They answer lies in his cardiovascular system. The Doctor has recently said what his limitations are when one heart isn't working at all. They sound very similar to our aged First Doctor. I can imagine a wound or disease that has damaged one heart (it would be the right one), leaving him with one good heart. Perhaps he's been told even a regeneration might not fix it. That plus his rebellious nature, and he steals a Tardis and runs off. But a damaged heart makes Hartnell's portrayal make complete sense.
The other big thing I have noticed is that Hartnell's early portrayal of the Doctor is that of your typical Timelord. He doesn't want to get involved, he wants to look and explore and leave. Yes, he is out running around Time and Space, but he is still a stodgy Timelord. But as time goes on, we see that early twinkle grow into the crusader he has become. And later when we see the Timelords, they are exactly what the Doctor was when he started. It was the adventures the Tardis took him on and the influence of Ian and Barbara that shaped him into the hero he is now. And this is all good character development in the first three years. It only got stronger when the Doctor became a younger, more energetic man and faced more deadly enemies. But it all started with a crotchety old Timelord who landed in a junkyard on Totters lane in London.
There is much to learn from these old episodes, both in what the producers and writers intended and how the actors approached their parts, that really enriches the tapestry that is Doctor Who. There are, of course, plenty of glitches that really don't make sense. Troughton's two later appearances in The Five Doctors and The Two Doctors don't fit with the continuity of his original appearance, and the first two Doctors helping to save Gallifrey in The Day of the Doctor last month doesn't fit with them being on the run. But these are minor points. They can drive you crazy if you let them, but if you ignore them you get to enjoy great TV drama.
There aren't any other TV shows I know of that have been so completely able to transition through different casts and production teams. Doctor Who has completely changed so many times that such changes are almost integral to its success. It has remained fresh and relevant for most of its run. But it really all goes back to the success of the First Doctor and it is fun to see how those early decisions have so shaped all that has come after and that it still fits. That you can believe that William Hartnell and Matt Smith are portraying the same character is a testament to just how much Hartnell's portrayal has shaped all the following ones.