I think what's missing in the discussion area is a bit more context. Taking all sets to failure would include the barbell sets. Taking your barbell sets to failure is not part of starting strength.
Why take body weight exercises to failure?
Body weight exercises don't cause near the stress on your body as the barbell work. Essentially, you can recover from them with more than enough time before you do them again. This approach increases the hypertrophy response and lets you get in some body image type training without spoiling the Starting Strength program.
It's my personal belief that what Rippetoe intended was what Wendler calls AMRAP (As Many Reps As Possible). AMRAP sets are until form starts breaking down, or you only have one more rep left in the tank. If you were to run that by Rippetoe, I'm sure he'd say something like "sounds good to me". He might have some wise cracks to throw in as well, but bottom line is that is in the spirit of what he wrote.
Bodybuilders who train to failure truly train a muscle group until it fails to produce any more force. The thought process is that it causes a greater hypertrophy response, although I'm not aware whether that has been proven scientifically or not. This level of training is definitely far beyond what Rippetoe intended. It can take over a week for those muscles to be recover and do that grueling work again. This is also definitely not what you want when you are training for performance as opposed to aesthetics.
Should I Train Pull Ups to Failure?
This is where you take a look at all the training work you are doing as a whole. If you are a climber, then the climbing work is a part of your overall program. If you already do pull-ups 3-4x per week, and you have decent volume (10-15 reps overall per session) then there is no reason to do additional pull ups over what you are already doing.
I do think that if you can hit 15 reps in a set, you do need to be adding some weight to make it more challenging. Just don't get in the thought process that you have to keep increasing weight all the time. As a climber I would work on adding weight until you have 10 lbs over the max amount of gear you are going to take as you climb. Then just keep adding reps.
The idea behind those changes are that you stay stronger than you need for the climb, and you build your endurance so you can keep doing what you need for the climb.