Basic satipatthana formula is
a monk dwells exertive, clearly aware, mindful,
observing the ____{body | feelings | mind | dhamma} in the ____{body | feelings | mind | dhamma},
removing covetousness and displeasure in regard to the world;
Mahā Sati’patthāna Sutta
Whatever Satipatthana you take you have to avoid covetousness and displeasure. Covertness and displeasure arises due to sensations which are pleasant and unpleasant. At the apparent level you might think you are attached or displeased with an external object. But in reality this is not the case. It is the sensation created with the object comes in contact with your sense door is what you get attached or averse to. You seek more and more contact which give the same sensation. With each contact one of the 121 mind states arises, which are all accompanied by sensation. You cannot be are or know anything without the sensation as any stimulus creates sensation which are part of the process of sensing the world.
For example say you are seated for a while. Body posture is part of the body contemplation. Being in a particular posture pain arises and you would want to move. When you move the new posure is pleasure. To introspect the body you need to feel around the body to know the posture. To do this you use your body sense door. It is contact on the body sense door and the sensations it create which allows you to know the body posture.
With regard to the mind, all metal state arise with sensation. (Sabbe dhamma vedana samosarana) So mind contemplation is very much tied to sensations also.
If you take the Dhamma contemplation, it is again the unsatisfactoriness of sensations due to their impermanence and dependently arisen nature link to the 1st Noble Truth. Sensations the fuel which makes the cycle to dependent origination go around. With regard to sense bases, contact give rise to sensations. So sensation is pivotal for this contemplation.
So whatever frame of reference you have to do it on what is felt, i.e., sensations / feelings.
When practicing the 4 Satipatthana what you should scrutinise is how is leads to understanding of the 4 Noble Truths, Dependent Arising, Displacing the Unwholesome Roots (Craving, Aversion and Ignorance), displacing the 5 Hindrances, developing 7 Factors of Enlightenment, etc. How the dots connect between the concepts and in case of the 7 Enlightenment Factors how to progress from one to another. All this tie ups with feeling, being equanimous and knowing arising and passing / impermanence.
Also you can look at it from the angle. When a Liberated Person is in Nirodha Samapatti (saññā-vedayita-nirodha), sensations and perceptions cases and metal processors halt. Also at time period does not experience Dukkha since the above 2 factors have ceased. A Liberated Person will experience the worldly experiences due to past Karma but will not create new Karma as he does not react to these sensations with craving or aversion. For this explanation let me take posture of sitting which is part of the Body Contemplation. If a Liberated Person is seated in Nirodha Samapatti no suffering nor unwholesomeness will arise from this though there is no awareness (no mental process). No suffering because there is no sensation and not unwholesomeness as there no reaction to sensations with craving or aversion. If a Liberated Person out of Nirodha Samapatti will still experience pain due to the posture after a while but there will be no unwholesomeness as you are not reacting to sensation with craving or aversion, hence no future result. A worldly person is also seated, fully are he is seated, but reacting with aversion to the pain due to the posture, which create fabrications and condition the future. So the key here is not reacting to sensations with wisdom (awareness of impermanence / arising and passing), even when doing any other Frame of Reference, which is the crux how S.N. Goenka teachers.
Also feelings (pleasant, unpleasant, neutral) is the one of the most recurring teaching or topic in the Tripitaka and in many contexts. E.g. antidote to sleepiness in Pacalā Sutta, letting go in Pahāna Sutta, 6 sense-bases in Sal-āyatana Vibhanga Sutta, Elements in Dhātu Vibhaṅga Sutta, Karma in Titth’ayatana Sutta, etc. This list goes on and on. Also the Buddha tailors the instructions based on the audience. For the perhaps the widest audience bases on the number of occurrence of where these instruction occur perhaps imply this is what works for the majority. Spread of the technique and the number of centres and number of people taking the courses also you can infer this also.
Also S. N. Goenka's teaching does not say focus solely on sensation. What is said is give importance to sensation.