It may well be that the only parts of Christianity that have transcendent value are the pagan ones, at least the parts left over after the Judeo-bits have been removed.
A deity that chose a favored tribe, even temporarily, just seems to be a poor choice for a universal force, even within his own postulated frame of reference. Not for those people fortunate to be enough in the chosen tribe, mind you; their choice is clear: just for everyone else.
But the Trinity may be an altogether deeper concept than the three hypostases of father, son, and holy ghost.
The neoplatonic intellect, soul, and spirit map even better than the orthodox Christian trinity to such common Christian triads as faith, hope, and charity, or word, love, and wisdom.
A trinitarian point of view has at least this plus: it adds complexity to the plane of moral or ethical values that compel a supra-naturalistic worldview, sufficient complexity to make religion viable without a necessary creator-god, or a presumptious classification of extra-physical forces. The abstract triads leave sufficient space for a life-giving holy spirit on top of physical mind and almost-physical soul that is universal and desirable to invoke and encourage in the microcosm, but not insecure enough to require worship or direct address in prayer.
As for why three should be such a satisfactory number, here's a simple explanation: three is the next ordinal after two, that is after the number essential to binary logic and thus empirical rationalism; it extends the binary in the most economical way.