As with all analogies about the Trinity, this one is flawed: it commits the heresy of partialism. The shell by itself is not fully the egg, neither is the white nor the yolk. Only all three components make up the egg. This is in contrast with the Trinity, where the Father is fully God, the Son is fully God, and the Spirit is fully God. The Father and Son are only differentiated from each other through the eternal relations of generation and filiation, and they are differentiated from the Spirit through the eternal relation of spiration.
It’s true that a tritheist could affirm this, but that doesn’t mean it’s unorthodox. What differentiates a polytheist from an orthodox Trinitarian is that the former will reject the additional statement that each of the three Persons fully embody the divine essence.
I agree. One important distinction (which I just learned) is between dogmatic and systematic theology. We are called to affirm the dogma of Christianity, which is embodied in Scripture and the historical creeds and confessions, even if we many not fully understand what is meant by them. The task of systematic theology is to make sense of dogma by connecting it to other dogma as well as other sources of information including general revelation. However, unlike dogmatic theology, systematic theology is not infallible and is continually subject to revision.
Another factor here is the doctrine of analogy. To some extent our understanding of God is severely limited by the fact that if God exists, then he is utterly different from and superior to anything we have encountered in the physical world. Therefore while we affirm what God has revealed to us in Scripture about himself, we are to be careful about taking the meanings too literally to fit our creaturely expectations. Everything we say positively about God is only true in an analogical, or approximate sense.