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The theological term for what you are describing is called Kenosis- from "an emptying." As you suggest, Philippians 2, in which the Scripture says that "though He was God, he thought equality with God was not something to be grasped," so he emptied himself and became obedient unto death, even the death on a Cross.
That God himself would choose to empty himself of the privileges of divinity is both a mystery (in that the mechanics and even motivation could be some what obscured) and a reminder of how much God loved us.
John 1 says that he chose to dwell among us - Exodus 33 says that man cannot see God and live. In order to reconcile these two facts, some accommodation necessarily had to be made.
Likewise, the true fullness of the glory of God is something that is simply so awesome, that had he not emptied himself, we would have necessarily been overhwhelmed. Like Henry IV who famously dressed up us a commoner so that he could go out and speak with his men as one of his men, so too did God necessarily become like men in order to be with men.
For this reason, Jesus' human nature was thus the dress in which his fully Divine nature was "pleased to dwell." He gave up the powers of God in order to be with us.
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Not sure why we insist that each of the persons of the trinity must incorporate the three omni (omniscience, omnipresence and omnipotence). Jesus is the eternal Son. Further Jesus claimed His work was dependent upon the omniscient will of God the Father (John 5:19, 5:30). Jesus however is credited with Creation (Colossians 1:16) thus He is omnipotent. The Holy Spirit is omnipresent. Clearly because they are a Tri-unity attributes are shared.