The Trinity is necessary because it stems from God’s very nature. Because of what God is, he must necessarily be Trinity.
The quick version. It’s because God has revealed himself as Knowledge, and he has revealed himself as Love.
Now, keeping in mind that there is no “before” and “after” in God, we must necessarily proceed in that manner because of our limitations of language and time.
“Before” anything else, God, being a Person, had to have an object of his Knowledge, but the only object of his Knowledge is himself. So God forms an Idea of himself. Now an Idea is a concept, and image, if you will, of the object. I have an idea of myself, and that idea is in a way, an image of myself. But my idea of myself, while like me, isn’t myself, because it is inadequate. I do not have the means or capability of forming a perfect idea of myself. But there is nothing inadequate in God, and his Idea of himself must itself be perfect, Adequate, infinite. The Idea is distinct from the Thinker, but the Idea must necessarily be infinite, perfect, and adequate. But there is only one infinite, and perfect. This Idea must necessarily also be God for God is incapable of forming an inadequate idea of himself. This Idea is the Logos. It is distinct from the Thinker, but just as infinite, perfect, and almighty. It is God, but also a distinct Person within the Godhead.
It must necessarily be, otherwise, God’s Knowledge is inadequate, and if there is any inadequacy, then he is not God.
Now, it must necessarily follow that these two Persons within the Godhead must share a bond of love with each other. We love too, but our love is necessarily limited by our own finite nature. Well, the Love shared between Father and Son cannot be less than infinite. It must necessarily be perfect and not lacking in anything. For this Love to be perfect and infinite, it would itself have to be Divine, and therefore, a Person. This is the Holy Spirit. It has to be, otherwise, we will have to claim there is something imperfect in the Godhead, and that is by definition impossible.
And again, while we have describe it in sequential terms, all this is taking place in an eternal Now.
As for Jesus in the Eucharist, he is not separate. Where one Person is, so must the others Persons necessarily be, because God is simple; he has no parts. The substance of the Eucharist is Christ of course, but where Christ is, the entire Trinity is there.