Trinity theories can be broadly categorized into two groups:
The above is an (over-)simplification of a topic which demands a lot of nuance to even begin to approach in a comprehensive or systematic way. See this entry in the Stanford Encyclopedia of Philosophy for a detailed breakdown of the various competing Trinity theories, which uses the terms "One-self" (Classical) or "Three-self" (Social) to refer to the broad categories discussed here.
From a Unitarian perspective, the main issue with the common forms of each theory is as follows:
These types of critiques have long been put forth by Unitarians, as well as Trinitarians of the opposing views. What follows is a Unitarian pointing this dilemma out as far back as 1833:
A Statement of Reasons For Not Believing the Doctrines of Trinitarians, by Andrews Norton - Section II, pg. 46
But in regard to all such accounts of the doctrine, it is an obvious remark, that the existence of the Son, and of the Spirit, is either necessary, or it is not. If their existence be necessary, we have then three beings necessarily existing, each possessing divine attributes; and consequently we have three Gods. If it be not necessary, but dependent on the will of the Father, then we say, that the distance is infinite between underived and independent existence, and derived and dependent; between the supremacy of God, the Father, and the subordination of beings who exist only through his will.
From the Unitarian point of view, then, both positions have major flaws, which seem to have no easy reconciliation.
What follows are Classical Trinitarians stating that Social Trinitarianism/Eternal Functional Subordinationism is Tritheism, or approaches Tritheism:
Simply Trinity, by Matthew Barrett - Chapter 5
Despite protests to the contrary, social trinitarianism has all the ingredients for tritheism. For where there are three wills there are three separate centers of consciousness, and where there are three separate centers of consciousness there are three separate gods. In this view, God no longer acts as one because he is one (inseparable operations), but he acts as one because the three wills of the three persons merely cooperate with one another.
Contradicting EFS, Social Trinitarianisms, and TriTheisms, by Bobby Grow
Part and parcel with eternal functional subordination is a social trinitarianism wherein we seemingly have three subjects, not one in the Divine Monarxia. I note a social trinitarianism inherent to the EFS position precisely at the point that we can ostensibly think that the Son could somehow be eternally obedient to the Father in a 'subordinate' manner. This reeks with the notion that there is a rupture between the Father and the Son (and thus by extrapolation, the Holy Spirit) such that the Son's being is distinct from the Father's; just at the point that it's conceivable that the Son could in any way be subordinate to the Father in the eternal reality. What we have then, if this is the case is a tri-theism; viz. the idea that there are three distinct centers of consciousness within the Godhead.
The New Catholic Encyclopedia - Volume 14, "Trinity, Holy"
This trinitarian concept of the divine Persons as three independent subjects (which owes much to Hegel) does away with any notion of the divine unity as numerical in favor of an organic unity that results from "the coworkings of the three divine subjects" (Moltmann). Clearly, this is a social model of the Trinity, which unavoidably runs the risk of tritheism. Father, Son, and Spirit are three "non-interchangeable subjects" whose unity is rooted, not in an identity of substance, but in a common history.
In response, what follows are Social Trinitarians stating that Classical Trinitarianism's procession doctrine terminates in Jesus or Jesus and the Spirit being inferior to the Father, or not fully divine themselves:
William Lane Craig - Is God the Son Begotten in His Divine Nature? TheoLogica 2019, Volume 3, Issue 1, pg. 26
God could simply exist eternally with His multiple cognitive faculties and capacities. This is, in my view, all for the better. For although credally affirmed, the doctrine of the generation of the Son (and the procession of the Spirit) is a relic of Logos Christology which finds virtually no warrant in the biblical text and introduces a subordinationism into the Godhead which anyone who affirms the full deity of Christ ought to find very troubling.
William Lane Craig - Is God the Son Begotten in His Divine Nature? TheoLogica 2019, Volume 3, Issue 1, pg. 27
This doctrine of the generation of the Logos from the Father cannot, despite assurances to the contrary, but diminish the status of the Son because He becomes an effect contingent upon the Father. Even if this eternal procession takes place necessarily and apart from the Father's will, the Son is less than the Father because the Father alone exists a se, whereas the Son exists through another (ab alio).
A New Systematic Theology of the Christian Faith, by Robert Reymond - Chapter 9
The only conclusion that one can fairly draw from this data is that Scripture provides little to no clear warrant for the speculation that the Nicene Fathers made the bedrock for the distinguishing properties of the Father and the Son. In fact, when they taught that the Father is the "source" (archē, or fons), "fountain" (pēgē) and "root" (rhiza) of the Son and that the Son in turn is God out of (ek) God, that is, he was begotten out of the being of the Father by a continuing act of begetting on the Father's part, they were, while not intending to do so, virtually denying to the Son the attribute of self-existence, an attribute essential to deity. There were exceptions among the Fathers, such as Cyril and the later Augustine, who did not teach so.
Trinities Podcast 363 - Dale Tuggy and Andrew Hollingsworth, timestamp 4:03-5:42
Tuggy: So Eternal generation and procession, you know, arguably they're very Central to "small c catholic" Christian theologies, but nowadays a lot of informed scholars think that these claims have no support in scripture, and sometimes they're also rejected because they imply that the Son and the Spirit lack the Divine attribute called Aseity. Do you agree with these two objections about no support in scripture - I think you just said you did agree with that - and what about the one that it implies basically ontological subordination for two of the three Persons?
Hollingsworth: I do tend to agree with these, and the move you typically see amongst contemporary philosophers and theologians is that they'll try to say something like this: Aseity is an attribute of Divine Essence. It's not a notional property. So while the father is ingenerate, it's acceptable that the Son is generate - that the personhood of the Son does not have Aseity, but the Son is still a se in the fact that it just has the Divine Essence. But I find this to be a very difficult claim because again on this classical view, the Persons are numerically identical to the Divine Essence. So it just doesn't make any sense to me to say well, the Father who is numerically identical to the Essence is ingenerate, the Son who's numerically identical to the Divine Essence is generate, is caused, so the Father in His personhood has Aseity, the Son in His Person lacks Aseity, so the Son somehow is a se in his Essence but not a se in His Person - but I don't know how to make sense of that if the Person is identical to the Divine Essence. That just seems like a clear contradiction to me.
Trinities Podcast 363 - Dale Tuggy and Andrew Hollingsworth, timestamp 10:42-11:37
Tuggy: Speaking of that - I mean the objection that generation and procession imply the subordination or the ontological lessness of the Son and Spirit, there is a traditionalist answer which you can find some scholars today defending which is from the Cappadocian fathers. They would say hey look, we just define that Divinity does not include Aseity - that's crazy Aryan stuff, we don't like that - and so we're just telling you that Divinity doesn't include Aseity. So, therefore, there's no problem with one Being passing on Divinity to another, and therefore existing because of the first. Why not that answer?
Hollingsworth: I guess because I think Aseity is just a pretty clear - even though it's not called "a se" - I think it's a clear Biblical teaching, that God is a necessary (Being), that God has always existed, and I think it's also including the idea of being a perfect Being. If God is a necessary Being, I don't see how He could be any other way but a se.
In conclusion, both of the major groupings of Trinity theories have primary disqualifying issues, which are picked up on in critiques by other Trinitarians who hold the alternative views. Attempts to defend against these critiques can get complicated rather quickly, which is demonstrative of one of the main arguments for Unitarianism, namely, that if this were something that God intended for us to believe, it would be explained in the text of Scripture by one of His prophets.