Listen carefully, because all examples are either Modalisms or Tritheisms and very clearly in all instances, conversations about Trinitarianism end the same way. That is: “mystery”.
If there were a more desperate word than “beg”, I would use it. It begs the question of how on earth such an incoherent or at least incomprehensible idea as the Trinity could play any role in Christianity, much less the central role it has since it was literally invented in the fourth century.
The Trinity cannot be explained.
The Trinity cannot be used to explain anything else as a result.
Why on earth do people think it matters?
Ultimately, it doesn’t tell us at all how God, Jesus, and the Holy Spirit are related. When we wish to talk about that relationship in terms of oneness, the polytheism required by the definition of the Trinity cancels out any relevant discussion. When we wish to speak of the distinctness of the persons of the Trinity, the modality required by the definition of the Trinity removes all ability of speaking that way.
Together, there is no hope of having any meaningful conversation about the Trinity aside from parsimony: that it was purely an invention of Rome, attempting to appease Jewish, true monotheism with Greek polytheism and articulated in a way such that the impossibility of such a mix is to appeal to a mystery that if not accepted, literally ended your time on earth, and usually by barbequecion.
About the Trinity, there’s nothing else relevant to say because it literally cannot mean anything as long as its definition remains a violation of the law of non contradiction. And as for its centrality in Christianity, it certainly is at least as useful as saying “Spot, Rover, and Benji are the same dog, but Spot, Rover, and Benji are also not the same dog.” Christianity, in 385 CE, put at the heart of itself an absurdity and somehow remains content to do so.
The only true mystery of the Trinity is why folks still believe in the idea, especially when no one can articulate what they’re even believing!