What happens when professional philosophers, trained with all their formidable analytic skills, turn their attention to an important Christian doctrine like the Trinity?

Things get tidy; radically tidy.

Philosophers do their philosopher thing, sorting things out, clarifying terms, making sure definitions are used consistently, criticizing bad arguments, and trying out new arguments. They disambiguate. They go formally logical. They reduce opponents ad absurdum, defeat defeaters of defeaters, and shine the glaring light of logical consistency on all claims –no matter how thick the fog of metaphor may be. They often try to be polite, but they don’t flinch from pointing out that you just said 2,000 words without actually making any claims which could be determined to be either true or false. They ask what counts as evidence, then generate new hypotheses and conceptual models to do justice to the evidence. After that, other philosophers come along and problematize those new hypotheses, and the cycle starts over again at a higher level.

The result is an enterprise called philosophical theology, and it’s a discipline that’s generated shelves of books and entire scholarly journals. There are hundreds and hundreds of professional practitioners and they have big academic conferences.
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