A first step in avoiding the paradoxes of the Trinity is to appreciate the word “trinity”. The -ity clitic, like “-ness” and “-hood” is a way of turning an adjective into an abstract noun, i.e it is a way of moving from a form in a subject (the red truck or personal car or a stupid question) to considering the form itself (redness, personality, stupidity, etc.).
The peculiar character of abstract nouns is to be both wholly present in individuals and yet wholly separate from them. It is the case both that you have your personality and I have mine, and that personality can continue to exist and be studied after any of us have died. Again, when you try to teach kids two from examples, you are hoping that they both see something in the examples and separate from them.
In other words, we’re used to there being a way of existing that is both entirely unified and distinct from the concrete while entirely present in diverse concrete realities. In calling God “trinity” we want to target exactly this sort of existence, though said of “three”. There will be defects in the description, but nowhere near as many defects as would be involved in calling God triple or threefold. It is straightforward polytheism to pray to the triple god, i.e. the god who was a whole composed from three persons as parts. But to pray to the threeness is not at all the same thing: threeness is clearly a unitary form, even while being wholly present in every triple.