Yes, while Rippetoe is a strength specialist (hence the name of the book containing "strength"). I disagree that it is not geared to other non pure strength athletes and sports. Improving strength is positively correlated with an improvement across all physical pursuits, so certainly is more broadly applicable to pretty much any human. There is no instance that I can think where an increase in strength (even if nominal) will not improve performance or quality of life.
It's not about being a Rippetoe disciple, its simply that there is a general void of excellent material by which people can self-teach themselves technique and generate an effective program. Wendler's 5-3-1 and much of what comes out of Westside are practically incomprehensible and not intended for anyone just getting started. My recommendation has nothing to do with my thinking that there is 'one-way', this is just the most readable, understandable and actionable resource available.
And I'll humbly disagree that the rocker board provides any additional benefit to the hips (barring any unusual injury) then simply getting stronger will have. I disagree for the same reason that I don't do any situps. What stress would a situp provide to someone that can 2x BW DL and squat 100+ of their BW? What deep muscles of the hips are not getting their share of the work when I squat 250# and all the balance and full range of motion required to do so? How is squatting my body weight on a wobble board providing any more beneficial stimulus that is not already surpassed by simply getting stronger under the bar? None. Zero. Zip. If I get my squat up to 300#'s the wobble board is not going to be a big deal. I'll never squat 300#'s on a wobble board.
yeah, i should have been more clear that I wasn't necessarily talking about the book but some of his other articles. i'm also not saying that the book isn't valuable or doesn't have cross-over benefit, but disagreeing with the pursuit of strength over all other factors. for most athletes that is not the way to train. besides starting strength, there is also the bill starr book and the even the bigger, faster, stronger book while geared to high school athletes is also a great resource for people just starting out.
the rocker board provides the benefit of having to work against an unstable surface thereby requiring a higher level of balance and stabilization than squatting on a flat, stable floor. the rocker board will force you to correct for instabilities that one may not even realize exist when only squatting on a stable surface. from an athletics point of view this is critical as most sports are not played in a controlled and perfectly stable environment.
for your situp example, while i do not promote situps for other reasons it is still a highly functional and necessary movement. so yes, squats and deadlifts provide crossover benefit for the core musculature, they do not replicate the movement of a situp.
squatting on rocker board will challenge your proprioreceptors more than squatting on the floor and that will yield benefit in terms of hip and trunk stability which will translate to improved athletic performance. rocker squats will also benefit your regular squat via the same improved stabilization. i haven't done them for a while as i've been doing other things, but i noticed an improvement on my regular squat when i was supplementing with rocker squats at load.
as for which muscles are getting more benefit - all of them. they are all having to work together to create the balance and stability required to do the squat. in theory, if you have excellent balance and muscular control you should be able to do a squat as smoothly on a rocker board as you can on the floor. if you can't then your musculature is not as mechanically efficient as it could be. i've experienced the difference it makes. before i started using it i had better than average balance doing a squat on the rocker board because of my squat strength, but after adding it in the rocker squats not only became better but so did the regular squats. i also assume you mean rocker board in the last sentence and not wobble board. however if you do mean wobble board then yes, it is going to be a big deal. using the wobble or rocker boards are learned skills that can't be mastered via the squat although they will be easeir to do than someone who does not squat at all.
i'm not trying to advocate only training on the rocker board, but it is a tool that has it's place and will add to your regular squat.