Well, there is a serious lack of discussion on what the Doctor's fear in the story actually.
There shouldn't need to be. Understanding a story isn't just about what the characters explicitly say, it's about recognizing what motivates it, considering the dialogue and actions in context with the events and the characterizations. In a well-told story, a character's actions and choices should reveal their motivations without anyone having to actually say the words "I am/he is motivated by this and that."
In this case, the Doctor does say that facing the Great Spider and bringing back the blue crystal is something he has to do, that there are things more important than going on living. If it were just about facing an icky spider, there'd be no reason he
had to do that. That would be strictly optional. The fear he had to face was clearly an urgent one, one that he couldn't avoid, one that he was obligated to face -- such as the fear of sacrificing his life to end a danger that his mistake had helped to create. So just from a very simple, straightforward story logic standpoint, it makes no sense to think it was about anything as superficial and random as arachnophobia. That's failing the most basic understanding that stories have logic and structure.
I don't think I've read that its about his death. In fact, Terrance Dicks himself dismissed the idea that the Doctor should fear anything, because he's fearless. To which I disagree, of course, because there's no point to a perfect hero, otherwise you'd get a comedy show, not the kind of show Doctor should be.
I don't know about that. The Doctor
is generally fearless, in the sense that he's extremely confident in himself to the point of arrogance. He trusts in his ability to win. So what he fears is having to face his own imperfection -- to admit that his mistake caused a problem, and to face the fact that he can't survive the act of fixing it.
Consider -- this was the first time the Doctor ever had to
choose to take an action that he knew would end his current life. His first regeneration was basically just from old age, and his second was forced on him by the Time Lords. He'd risked his life before, but he'd always been able to think his way out and not have to pay the ultimate price. This time, there was no way out. He had to go into danger that he knew for a fact would be fatal -- and he couldn't be certain he'd be able to regenerate, given the severity of the damage he'd sustain (indeed, without Cho-Je's help, he wouldn't have regenerated). This was his
Kobayashi Maru, the no-win scenario he'd never faced before. That was his greatest fear: having to accept that there was a problem he couldn't think his way out of, a situation he could not control.
Of course, it's worth noting that it wasn't the last time he chose to sacrifice himself. Once he'd faced his fear the first time, it made him able to do the same again -- first to sacrifice himself to save the universe in "Logopolis" (though he knew he'd live on that time, thanks to the Watcher), then to sacrifice himself just to save a single person in "The Caves of Androzani," a far greater act of selflessness. And every one of his regenerations from "The Parting of the Ways" onward has been a similar act of knowing self-sacrifice, although in "The Time of the Doctor" it was a sacrifice that took about 900 years to pay off.