Agreed, the Western church has been guilty of a non-Trinitarian approach to things at times and in places. Each of our churches has been imperfect in some regard or another, and even when we honour the Trinity with our mouths, we sometimes do not honour Him fully with our hearts. The work of the Holy Spirit in particular come to mind.
The triple baptism though, as your texts clearly show, are a human tradition. Scripturally, we are not told that Christ/God/Spirit requires it.
“You were conducted to a bath just as Christ was carried to the grave and were thrice immersed to signify the three days of His burial.” (Clement of Alexandria)
Note the word ‘signify’. We should be careful to separate those things which are tradition (even beneficial tradition), and those things which are for salvation.
The tradition is a lovely one, and many ‘Western’ churches use it as well... it is not however a requirement for salvation.
So, when you form your religious opinions based on English translation you are reading "I baptize you..." which doesn't even describe what that means, let alone the manner of such act.
And Greek reads the same passage and sees "I immerse you repeatedly ... " and the two of you do not read one and the same thing. And that carries a completely different meaning. This is also an excellent example of how one's theology cannot be formed properly by simply reading any version of the Bible because all translations are corrupt. Thus, to a Protestant who makes up his or her own rules based on reading some Bible, there is nothing wrong with just immersing once. In fact, more than once becomes a 'tradition of men' quite erroneously based on erroneous reading. Once the error is set, errors proliferate.
Now, I will agree that thrice is probably a tradition that the Church adopted as the meaning of baptiso based on the trinitarian concepts, God's three days of burial before resurrection, etc. quite in line with Gospels' witness and not as something arbitrary or, worse, as a result of misunderstanding.