As best I can tell, the book of Acts, especially Acts 2, provides one of the most important "centers" or controlling elements in the way this group reads Scripture. Thus Acts 2:38 says, "Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of sins, and you will receive the Holy Spirit."
The group correctly reads the verse to say:
1. Peter associates the forgiveness of his audience's sins with baptism.
2. This baptism is "in the name of Jesus Christ"--nothing is said of God the Father or the Holy Spirit.
3. A person receives the Holy Spirit in association with such repentance and baptism.
Let's see how the group relates these three aspects of the verse to their key beliefs.
First let's look at their insistence that such baptism be in the name of Jesus [only] and their rejection of the idea of the Trinity.
They have correctly noticed that the book of Acts consistently involves a baptism in the name of Jesus. Nothing is said in Acts about baptism in the name of the Father, Son, and Holy Spirit. Indeed, I suspect they are right to think that the earliest Christians baptized in the name of Jesus (cf. Rom. 6:3).
But as pre-modern readers, they cannot allow other segments of the early church to baptize in the name of the Father and of the Son and of the Holy Spirit (e.g., Matt. 28:19). They view the Bible as a single text written by God, so the historical tensions that almost certainly existed in the early church are mowed down by the lawn mower called "harmonization."
At first glance, harmonization looks like it is the most godly approach to Scripture--finding ways to reinterpret passages of the Bible so that their meaning is no longer in tension with the meaning of other passages. When the Holy Spirit is truly behind such harmonization (that is, leading the reader to truth through it, even though the truth is different from the original meaning) or when a person is either consciously or subconsciously doing it with orthodox faith setting the boundaries, I don't have a problem with it.
But I have two problems with it when it is used to support unorthodox views or when individuals (I am not referring to this group now) use harmonization to dismiss sincere Bible scholars as unspiritual because they don't harmonize. The first problem I have is that harmonizing ultimately rejects the voices of Scripture and substitutes itself for Scripture.
For example, Matthew and Mark record Jesus' healing of a leper and his healing of Peter's mother-in-law in opposite order. A harmonizer might suggest Jesus healed two lepers: one before Peter's mother-in-law and one afterwards. This example is harmless enough, but you have created a senario that neither of the gospels present. You have invented your own scheme and, in a sense, rejected both what Matthew and Mark actually say.
Again, in this case the result is really not that big of a deal. But the Jesus only group actually rejects Matthew's baptism in the name of the Father and of the Son and of the Holy Spirit in favor of Acts. A clever rationalization ensues--and such splicing is often ingenious. They might argue that "name" is singular and invoke some interpretation that says these three are really all Jesus, just in three phases of his existence. What might seem to be a baptism in the names of three different persons is really just baptism in the one name of Jesus.
This leads me to the second problem I ultimately have with harmonization when it becomes unorthodox or is used to support a condescending attitude toward Bible scholars (again, I am not referring to the group with this last comment). The resultant interpretations are simply wrong in terms of what the text originally meant. It's playing a game with the text. A Bible scholar who genuinely sees tensions in the biblical text may have a much greater respect for the text of the Bible than a harmonizer. That's because such a person is willing to let each part mean whatever it seems to mean. In other words, such a person doesn't shove harmonizations down the Bible's throat. Some people's "idea" of the Bible is far more important to them than the text of the Bible itself.
So in the case of the leper and Peter's mother in law, the wording of the leper story is extremely similar in both gospels. Indeed, it is overwhelmingly likely that Matthew and Mark stand in some literary relationship to one another--that one is either copying its basic text from the other or that they are both drawing on a common source of some type. As such it is overwhelmingly likely that the leper story is the same story placed in different locations by the two authors. Harmonization in this instance is not only not important for faith but it is simply an incorrect interpretation.
Similarly, while Matthew uses the singular for "name," one can only support the idea that this is simply naming Jesus three times if one brings this pre-conception to the text. There is certainly nothing in the text that would lead us in this direction. The resultant interpretation may preserve the "idea" of harmony, but it does so by raping the biblical text of its actual meaning. It ignores what the text wants to say in order to preserve what the interpreter wants the text to say.
Indeed, if anything the New Testament itself pushes us to distinguish Jesus from God, not to identify them. Thus Jesus says he does not know things that the Father does (Mark 13:32). The Father speaks while Jesus prays (e.g., Matt. 3:17). Jesus is the firstborn of creation (Col. 1:15), which seems to place him on the creation side of the equation in Paul's mind (a situation those of us who believe in the Trinity also need to explain). The group no doubt has ingenious ways of reinterpreting these passages, as Christians in general have great "coping strategies" to explain away "naughty verses" that don't fit with our Wesleyan or Baptist or Presbyterian paradigms.
But when we really value the text (and are looking for the original meaning rather than a "spiritual" one), we don't find possible ways to interpret it so it fits with our theology, we try to go with the most probable reading of the text. Inevitably, we begin to see the Bible in context as a chorus of voices that basically are in harmony with each other, but we also see tensions between real people. From my perspective, you simply cannot let the Bible set the agenda without reaching some conclusion of this sort. To do otherwise is to foist on the Bible a view it does not "want" you to have. It would be like trying to honor me for being such an incredible war hero despite my continued protests that I have never been in a war or even in the armed forces. What I really want you to do is to listen to what I'm actually trying to say to you.
Now in saying such things I do not mean to negate my view that it is ultimately great to hear the Spirit speaking in the words apart from what they meant originally. What is important to me is that we don't confuse such spiritual meanings with the original meaning. And even more important to me is that we don't mistake the splicings of our own making with the authoritative voice of God.