There is no question that the current state of what I call the "church visible" (i.e., "official" Christianity organized into denominations) is largely spiritually moribund at best and apostate at worst. Further, the religious church is indeed prophesied to continue to deteriorate in the end times until, during the Tribulation, apparently all of what is now organized will be subsumed into antichrist's tribulational anti-Church (see the link: "The situation of the church visible on the eve of the Tribulation" and "The persuasiveness of the tribulational false religion" both from CT 3A).
On the Trinity, it would be best to consider first Matthew 28:19. In that passage we agree that the baptism is that of the Spirit. Furthermore, the Greek word used there for "Name", onoma, can and often does have the sense of "person" (i.e., for which the name stands), and clearly does so here. So it is the baptism of the Spirit which places the believer "into union" with the Persons of . . . . . the Father, the Son, and the Holy Spirit – therefore clearly and obviously the Spirit is a Person of God to the extent that and in the same way that this is true of the Father and of the Son.
Now there are many other passages which may be adduced from a positive point of view that show clearly the three Persons of the Trinity at once (see the links: "The One True God and the Trinity in the Old Testament", and "The Trinity in Isaiah 63:10-15"). But since a good deal of the justification for questioning the Trinity in this theory to which you refer seems to be based upon the numerous passages where the Spirit does not appear, it is important to point out that this is hardly evidence that the Spirit is not a co-equal member of the God-head. That would not be a logical deduction in light of the other passages where the Three are mentioned together and also the many places where the Spirit is mentioned either individually or in tandem with either the Father or the Son (see the link: "Questioning the Trinity").
It is true that the Spirit is the least mentioned of the Three in scripture. But to use that as an argument against His divinity is also to necessarily question the divinity of the Son. That is so because the Son, of course, is mentioned much less often than the Father because of the fact that His deity is largely shrouded in the Old Testament. In truth, many of the instances where the Father seems to be in view are actually instances of the Son acting in the Father's place, something that was not meant to be fully understood until Jesus was revealed in the flesh (see the links: "Jesus Christ in the Old Testament [Christophany: Gen.3:8]" and "Christophany in the Exodus"). But the fact remains that, to the casual reader of the Old Testament with no knowledge of the New, the divinity of the Jesus the Messiah might possibly be overlooked (especially if certain key passages like Isaiah 52-53 and the Messianic Psalms are omitted either out of ignorance or arrogance) – it certainly has been by the Jewish faith.
Doubting the divinity of the Spirit because of the relative paucity with which the Bible mentions Him is poor theology - the Bible only has to say something once for it to be true, and the divinity of the Spirit is established many times (as in Matt.28:19 and in the links given above). And there is of course good reason for this relative numerical scarcity, namely, it is the Spirit's role to warm, empower, inspire, guide, restrain invisibly. That is why Jesus says of the Spirit in the other passage we have spent some time on, John 3:3-15, that "the wind blows wherever it pleases. You hear its sound, but you cannot tell where it comes from or where it is going. So it is with everyone born of the Spirit" (Jn.3:8). One other important set of passages to consider are the visions in Revelation where the Father on the throne and the Lamb are clearly seen, but the Spirit, while mentioned prominently, is represented somewhat less visibly by "seven spirits", that is, the Spirit in perfection as the number seven represents Him (Rev.1:4; 3:1; 4:5; 5:6).
Many of the questions people have in this regard seem to be based upon the supposition that because the Spirit is sometimes called "the Spirit of Christ" or "the Spirit of Jesus", that these and therefore all other instances are to be taken as a periphrasis for "Christ/Jesus". There is no convincing argument to be made for this extension of the premise on the one hand, and, on the other hand, it is far easier and more natural to understand the Spirit as being "of Jesus/Christ" because it is Jesus Christ who sends Him and because He then teaches about Jesus Christ (e.g., Jn.15:26). This last passage, by the way, John 15:26, talks about the Paraclete, the Son and the Father in a way that certainly makes it seem like they are three distinct Persons (indeed, one would have to be operating on the "two Person theory" at the time when one reads this passage for any other interpretation even to come into the mind, a sure sign of trouble). The same is true of many of the other passages which are sometimes adduced as support for "two" but really have "three", such as 1Cor.12:4-6 where the Spirit, Son, and Father all have separate duties regarding spiritual gifts: the Spirit gives the gift itself, the Son gives the ministry or ministry field, and the Father gives the results of using the gift in that field.
One of the greatest challenges we face as believers on the cusp of the Tribulation, with so much of the church visible already having at least one foot in apostasy and the rest largely proceeding in the same direction, is to find a way to keep false teaching at arms' length while at the same time continuing to grow spiritually through attention to genuinely orthodox (i.e., "true and straight") teaching. Every believer is individually responsible for the choices he/she makes in this regard, and they are of no little moment. I am certainly prepared to commend good work when I see it, but also to censure what I believe to be wrong, especially when it is so obviously and dangerously wrong as this "two Person" theory is. I believe in the Three Persons of the Godhead not because of tradition but because that is what the scripture teaches very clearly in my view – just as a teaching is not necessarily true because it has the authority of tradition, so also the authority of tradition does not necessarily make it false. I see nothing at all in the examples and arguments often presented which I find compelling in the least. One of the dangers in being "iconoclastic" is that sometimes one gets so carried away with busting up "icons" that one takes the hammer to what is right and true as well. I believe that this "two Person" theory is a case in point, and I would recommend that all genuine believers give it a wide birth.
In the Name of Him who died for our sins and through His death gave us eternal life, our Savior Jesus Christ.