That’s an interesting line of thought; but I think readers in Paul’s audience would have been more familiar with Judaism’s emphasis on the number 7 (as in the 7 days of creation) or Pythagoras’ theory of number which emphasized all the numbers up to 10. (I’m not saying Paul followed Pythagoras, just that this was an intellectual framework that would have been familiar to his readers if they were thinking of the universe in terms of number.) Personally I think all numbers reflect God’s glory in one way or another. For instance God is One; the image of God is created male and female in Genesis; we have the Trinity and other instances of three as you mention; the four four-faced cherubim seen by Ezekiel; the twelve tribes of Israel; the 144,000 servants of God in Revelation; etc. George Ferguson’s “Signs and Symbols in Christan Art” has a bit on Christian number symbolism in Chapter 11 that’s an interesting read.
Man is three in one. Body, soul, and spirit.
When man sinned, on that day, he surely did die (in spirit).
The holy spirit comes to us and inhabits us today, in which we receive regeneration.
Matter=liquid,gas,solid
Elements=Neutron,Proton,Electrons
Earth=land,water,sky
Life=plants,animals,mankind
Space=height,length,depth
Time=past,present,future
Electricity=negative,positive,neutral
There are innumerable examples.
God exists as 3 in 1, just like Man does...we are an image of God. Of course ours is a very very poor image.