The classic Doctor doesn’t apologize quickly… except for at the end of episode 2 and beginning of episode 3 where he apologizes for them being captured by a nomadic tribe a hundred thousand years in the past. Susan then gives the most inane response ever, “It’s not your fault, Grandfather.” Except… it was!
Barbara and Ian has gone from being kidnapped by a smug time traveling alien to this tribe and death looming unless they can make fire for the tribe. The Doctor was over promising what he could do for the tribe and not caring at all about potentially changing the course of history. His level or self preservation was his undoing; leading to their capture and confinement in “The Cave of Skulls.”
Barbara saw this selfishness in The Doctor. When she questions why The Doctor is willing to help Ian and Barbara escape (despite the fact he originally wanted to drive them far off so they wouldn’t spill his secret,) he responds with another now famous line, “Fear makes companions of us all.” It’s the most unselfish thing he will do in the whole serials. Moffat would take this throw away, quick response, and weave The Who episode “Listen” around it. And, yes, I still cry every time a watch it, despite the fact that Gallifrey should have still be time locked… but, whatever!
However, this isn’t some magically turning point. The selfishness of the First Doctor is extremely dynamic and unique. It’s something that the curmudgeon Doctors (Six, Twelve) have tried to duplicate but can’t. Probably because it wasn’t an old man response; instead it was the response of a self absorbed, spoiled young Gallifreyan who lacked real experience in the universe.
After all, it is Barbara who convinces the group they must help an injured nomad, after they escape. Not the Doctor. It is Ian who takes the lead on treating him. Not the Doctor. It is Susan who provides assistance. Not the Doctor. In fact his response is that they must be out of their mind. He argues that their desire to be humane will be their undoing. He is not THE Doctor, yet.