Three “persons” = three individuals = three Gods. Maybe it is a mystery only because we made it that way.
If Jesus as the Second Person has always existed, for what purpose did He exist prior to His incarnation? Did the Holy Spirit always exist? Which was “first”? If they have one nature and one substance (if indeed God does have substance) then are we not talking about One God who appears to us and communicates in various ways, including via the Incarnation?
Maybe I am hung up on the term “persons”, which makes it sound like there are three distinct individuals.
Is it a mystery because God did not give us the wherewithal to understand or is it a mystery because we turned it into one? The OP asks about understanding the Trinity, and the apparent answer is we can’t. But the Church seems to have reasons and explanations for just about everything else.
Think about the “Lord’s Prayer”, the one that Jesus gave us. One God, no mention of a Triune God. Then look at the Creed, invented by people, with “I believe in One God” as the start, but then we say we believe in two others who are also God.
So if someone said that they believed in ONE God who could and has manifested Himself in different ways (Jesus, Holy Spirit) that would be wrong. But to say we believe in THREE Gods but really only mean one, that is supposed to be right. Hmmmmm.
Scripture (John 1:1) applies the term “Theos” to the Son. When it does this, Scripture creates a major problem for monotheists, who believe that there is only one God. Scripture previously has been adamant that there is only one true God; there were false gods, yes, but only one true God. But now in John 1:1 we have Scripture using the word Theos for a person other than the Father. The Father is obviously God, but now we have Jesus, the Son, the Word, also being called God! Quite a paradox. How to deal with this and preserve monotheism?
One early solution was used by the Docetists, that Jesus was the Father. Jesus and the Father were the same person. Hence, still only one God. Jesus only appeared to have come in the flesh, suffered and died. It was all an illusion.
However, Scripture is so clear that the Father and Jesus are separate persons, that the Docetic solution can’t be right.
Another solution (used by the Jehovah’s Witnesses) is that Jesus is “a god.” Not God, but a god. But this solution doesn’t really preserve monotheism either, because it sets up Jesus as a little God beside the true God. And since there is only one true God, the god Jesus must be a false one. The idea that Jesus is false is understandably rejected by Christians.
The idea that the Father, Son, and Holy Spirit are manifestations of the one God, is called “Modalism.” This too was rejected by early Christians again because Scripture definitely treats the three as separate persons.
So, the way the Trinity is defined is a balancing act, giving all verses of Scripture their weight about what the Godhead is. It’s true the “Trinity” is man-made in a sense, but it is the conclusion the Church is forced to by considering all of Scripture and Tradition.