Good AM...I love you and hope you are doing well!
All of those are intuitively appealing and have support from the Scriptures. But there are many verses which go further-- indicating something more than simple monotheism (e.g., the plural of Gen 1's "let us man in our image") and claiming that Jesus and Holy Spirit are deity (most notably, Jesus Himself claims deity, leading to the famous Lord/Liar/Lunatic/Legend dilemma).
More broadly-- and this doesn't prove anything, but is still important to consider-- there is always a tendency to deal with our inability to understand an infinite God, by reducing Him in some way to something simpler that we can fully understand. Unfortunately, when we simplify too much, in error, the result is not simple, but simplistic (and heretical). It doesn't have to be the case on this doctrine-- although I'm hard-pressed to imagine a better example (perhaps free will / predestination?)-- but in what areas is one's theology unable to fully comprehend an infinite God? If the answer is none or rarely so, then the odds increase that one has drifted into heresy.
One more thought-- on the practical implications of the Trinity as doctrine. If God is Himself "in community", it underlines the importance of community for our well-being. Again, this is not a proof, but the doctrine lines up with the emphasis on community throughout the Scriptures-- and encourages us to embrace the sort of community which is pressed as crucial throughout the NT.
Finally, have you read Chesterton's paragraph on the paradoxes of the faith?
