Since most books are subjective, I doubt there can be found an objective book on how vipassana of U. Mahasi and S.N. Goenka works. These methods are merely constructed out of theories. For example, the Buddha taught all conditioned things are impermanent. If you watch leaves falling from a tree or waves lapping on the seashore, you will see impermanence. However, this does not mean watching falling leaves or lapping waves will bring enlightenment. Similarly, just because U. Mahasi and S.N. Goenka have chosen impermanent meditation objects, it does not mean these methods alone will work.
In the Pali suttas, it is reported the Buddha taught meditation as the development (bhavana) of concentration (samadhi). Properly developed samadhi has two fruits or results, namely, samatha (tranquility) and vipassana (clear seeing of higher truth). If the mind does not have samadhi (which itself is comprised of a degree of selflessness or mental purity) then watching the impermanence of falling leaves, lapping waves, abdomen rising & falling or painful sensations probably will not result in sufficient vipassana to end the defilements and liberate the mind.
Thus, to answer your questions:
Noting and paying attention to the movement of the stomach is theoretically supposed to create understanding because the mind is supposed to see constant impermanence. If the mind can see every expansion of the stomach and every contraction of the stomach then it can see things (such as the stomach, breathing & mind consciousness itself) only last for a moment. If such clear seeing of impermanence occurs, then the mind will know all things, including life itself, are impermanent. The mind will be peaceful & calm when impermanence (such as loss of property, loved ones or life) occurs. But, as I previously said, if the mind has too much ego, the meditation really won't be effective because there is too much ego in the mind. The mind is too impure. The seeing of impermanence is not clear enough and not strong enough.
Not reacting to sensations throughout the body theoretically leads to insight because two things occur: (i) concentration is developed via non-reacting; and (ii) since reacting is disturbing, non-peaceful or suffering, by non-reacting, the mind can have the insight that non-reacting is peaceful. However, if samatha (tranquility) and selflessness has not been developed then there will not be much peace to be felt via non-reacting. In summary, the Buddha's path is comprised of eight factors. Each of the eight factors must work together to get the result of liberation & peace.
Similar to the above, restraint of the senses and mindfulness leads to liberation because the mind is not stirring up reactions and defilements that are suffering. However, the mind must have some peace before restraint can lead to liberation. If the mind has many hindrances & defilements, trying to restrain the mind will be like trying to restrain a tidal wave or tsunami.
In conclusion, the Buddha's path is the Noble Eightfold Path. The whole path must be practised, which includes wisdom-based-morality and samadhi. Trying to follow moral rules without wisdom about them will also not work. Trying to develop samadhi without wisdom also will not work. Trying to develop wisdom alone also will not work.
Vipassana is one of the final results of the path, similar to the top of a tree. A tree is not climbed from the top. A tree is climbed from the bottom. Similarly, attempting to begin meditation with vipassana probably won't work. Instead, we should try to make the mind calm by learning how to end hindrances and sit at ease with awareness of breathing. "Vipassana" (knowing & seeing things as they actually are) is something that occurs without any act of will. You may refer to Cetana Sutta: An Act of Will.