Well, that was surprising. Jacob, the TWoP recapper for Doctor Who, got in touch with me about my problems with his D&D reference. Apparently I was overthinking the joke- what a shocker!
At first Jacob explained that he was using the d20 roll to imitate the non-player-character behavior of the TARDIS against the Doctor. I replied, asking if the Doctor has to make a diplomacy check every time he wants to travel somewhere, or whether it was more like a will save that the TARDIS had to make in order to be influenced/controlled by the Doctor. Jacob agreed that it was more like a will save, but that on its simplest level, the joke was merely that there were 20 things that could go wrong with the TARDIS, and that each time the Doctor was making a roll to see which outcome he’d get.
This entry is really just so I can end with this testament to my nerdliness and obsessive nitpickery. “I just didn’t think anybody would catch the reference in the first place, so it bugs me that it didn’t make sense. I should have worded it better.” Yaaaaay.