The deeper reason for requiring us to fast on Shabbat/Yom Kippur may lie in the essential difference between Shabbat and Yom Kippur. Shabbat is God’s day of rest. Yom Kippur is our day of rest. On Shabbat we remain in this world and enjoy the earthly pleasures God created. That is how we acknowledge Him as Creator. Shabbat in this sense is a day of physical rest.
On Yom Kippur, we leave the physical world. We take a day off from the inherent tension caused by the fusion of body and soul. We become pure souls. Yom Kippur is the dress rehearsal of our own yahrzeit. We wear the shrouds in which we will ultimately face Him, and we discard the shoes we will no longer need. By neither eating nor drinking, we celebrate the day like the angels we become. If Shabbat is God’s day off and Yom Kippur is ours, on Shabbat/Yom Kippur God and His people celebrate a day off together.