There are a couple major issues here. 1. How ought we to understand the use of terms like 'theos' in early texts such as St. Ignatius of Antioch's? 2. What were the beliefs of the early church, and how did those change leading upto the official adoption of Trinitarianism, which happens hundreds of years later? These two questions are obviously inter-related, where our understanding of each will impact the other.
Behind this is the ambiguity of the phrase 'divinity of Christ', used by the OP. What does it mean? Does it include Unitarians like Arius, who held Jesus was 'theos' but not co-equal with the Father? Does it include Unitarians who hold Jesus is at the right hand of the Father and given all authority in Heaven? I would say yes. Some would say no. So the question is ambiguous.
The first thing to note is that terms like 'theos' (Greek) and 'elohim' (Hebrew) didn't map neatly on to our English term 'God'. The words were more flexible, and could be used of heroes, angels, 'gods', or God Himself.
This leads to another point, which is that the ancient Jewish understanding of sender and sent was different than ours. Agent-identity (sender and sent) was closer linguistically to what we think of as ontological-identity (two names for the same being). This is reflected in the general use of the term 'elohim' ('God'), which is applied to heroes, kings, judges, or angels, in addition to God Himself. Why? These figures, as far as they were 'elohim', stood in an agent-relationship with God.
For an example of this kind of conceptualization in the NT, consider where in 2 Gospels a centurion has messengers speaking, but in 1 Gospel it is the centurion himself speaking. Well, which is it? To the ancient Jewish mind, the distinction was not as great as to our mind, because the language was just being used functionally, and the point in this case was the centurion's intentions - whether identified as being spoken proximally by a messenger or the centurion didn't really matter.
Not surprisingly, we don't have to go to St. Ignatius of Antioch (or later writers) to get 'theos' used apparently to describe Jesus. The main source for this is the Gospel of John, in particular John 1:1 (Word = Jesus, Word = theos) and John 20:28 (Thomas says to Jesus "my Lord and my theos"). You also have at least one other place where this is arguable (Hebrews), although we have issues of textual variants and grammatical ambiguity in (almost?) all cases.
With that prelude, let's talk about what happened. The first is a matter of history. The early Christian church, as far as we can tell with surviving theological works, was overwhelmingly Unitarian up until around the time of Arius (and then significantly Unitarian straight through to the 7th century, when Spain's King converted to Trinitarianism - so > 600 years of Unitarian tradition in the 'early Church', and then re-emerged very rapidly after greater theological freedoms with the Protestant Reformation, such that there is ~1,150 years of Unitarian tradition within Christendom). Arius, himself a Unitarian, claimed to simply be passing on received tradition, and this is plausible because of the theological record we have.
Jesus was a 'theos' to many of these Unitarians, because they believed He was a pre-existent logos being, a 'god', who was not co-equal with the Father but above a 'mere man'. This is very important. The works of various early Church theologians in the first 200 years after Christ, such as those quoted in the OP, when inspected carefully, typically don't show Binitarianism (or Trinitarianism), but logos-theory Unitarianism or something similar. 'Theos' is an ambiguous term whose meaning in early Church writing must be determined by context, not proof-texts (as is also the case with the NT and OT!).
So the real question for a contemporary Biblical Unitarian, who holds Jesus is entirely a man and doesn't pre-exist - is why these early Unitarians, such as logos-theorists, went wrong and went wrong so quickly (it's only later on that these errors lead to further errors with Binitarianism and then Trinitarianism).
This is confused by a couple issues. The first is that saying Jesus is 'theos' or 'elohim' (as Moses is described, Ex. 4:16, 7:1) could mean the author is claiming an agent-role, not an ontological-role. Plausibly for BUs, this is what Thomas is doing at John 20:28, recognizing that Jesus is indeed the agent of the Father and that, therefore, when he sees Jesus he sees the Father.
The second is that 'theos' or 'elohim' can be applied not just for a man who is an agent, but also an angel or logos-being.
The first sort of interpretation of early Church writers wouldn't actually be an error for BUs. So the question is really why logos-theorists, who held to Jesus being something like an angel (similar to JWs nowadays) or logos-being, developed in the early Church - and indeed, we know this happened quickly.
The standard answer is the explosion of theologians in the Church who simultaneously were strongly influenced by Greek philosophy, and who were weaker at understanding ancient Jewish ways of thinking. It was this one-two punch which knocked the early Church off its course, and launched the logos-theorists.
So what happened with these early Unitarians, who held not only that Jesus was 'theos' (a BU could hold the same thing in the proper sense), but that Jesus was a pre-existent logos-being?
The answer is a) they didn't understand the ancient Jewish idea of ideal pre-existence properly (in particular, that the name of the Christ or Messiah pre-existed before the foundation of the world as an idea in God's mind), b) they misunderstood the strong idea of agency that was reflected in John's Gospel (exemplified by John 20:28), and c) they imported a philosophical preconception which required an intermediary between the Father and the world in order to retain the Father's 'purity' from the world but enable the Father to act in the world, i.e., a 'Son' or 'Logos' figure. These 3 strands combined to form Unitarian logos-theorists, who sprung up very quickly once the Apostles died and an explosion of Greek philosophy-informed Christian theologians arrived in the early Church.
So, a) when Jesus speaks of the glory He had with the Father before the foundation of the world (John 17:5), instead of seeing this as a statement of ideal pre-existence of the name of the Messiah (as in the Babylonian Talmud), they thought it was a claim by Jesus of literal pre-existence.
This is a talk by Bill Schlegel where he articulates the basic BU view of John 17:5, including ideal pre-existence and the prophetic past in ancient Hebrew.
When b) Thomas says to Jesus "my Lord and my God", being unfamiliar with the Jewish idea of agency, they saw this as Thomas literally claiming Jesus is a god (again, the logos-being who pre-exists and is a go-between), instead of seeing it as agency talk, with Thomas recognizing that Jesus is the agent of the Father, i.e., of God.
When c) the narrator starts talking of a 'logos' through whom 'all' come to be in John 1, they saw this in terms of a 'demiurge' or go-between, a lesser deity in Greek philosophy, and so the logos is 'a god'.
For c) here, BU views about John 1's prologue differ, so views about which exact error was made by early Church logos-theorists will differ. Some, like John Schoenheit, hold the logos is a 'plan' here. Others, like Dale Tuggy, hold the logos is a personification of God's creative power (similar to the personification of Wisdom in other parts of the Bible). Finally, others, such as Bill Schlegel, hold the logos is a title for Jesus (and I fit in this last category) and so are closer to typical Trinitarian views, but that the 'beginning' is the new beginning of Jesus' ministry (so, Socinian) and so doesn't speak to a pre-existent logos-being. So, to some extent, views on what went wrong in the early Church with John 1 are going to be indexed to different views on what the correct interpretation of John's prologue is, and so will vary amongst BUs.
However, all BUs hold that the broad swath of scripture teaches that Jesus is wholly man, and did not pre-exist as some sort of other being, and so will agree that John 1 is a key place where people in the early Church veered into error, leading to logos-theorists -> Binitarians -> Trinitarians. From the BU perspective, the original sin here is not Trinitarianism, but 'Jesus is a god' Unitarianism. It is anachronistic according to standard BU views to think a 2nd century writer was thinking in Trinitarian categories.
