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I don't believe there is a standard unitarian Christian way to understand or respond to this specific argument. However, I can point out some potential problems with it that I am guessing other unitarian Christians would also point out.
One problem is with terminology. We have God in a general sense in Trinitarian theology, which includes all 3 persons of the Trinity. Then, we have God in a narrower sense of 1 person of the Trinity - in this sense, all 3 persons aren't included in the term 'God'.
So it is probably helpful to be a bit clearer about our meaning in the argument. For example, God in the general sense has no one else to love (if we're not taking creation into account) even if there is a Trinity, but the Father (in the Trinitarian sense) has someone else to love if there is an eternally begotten Son.
So the argument really has to be that having an 'internal' relation of loving persons is a good property, and that property is lacking in the unitarian conception of God, and therefore the unitarian God isn't all-good.
(This is different from the argument in the video, which trades on the distinction between finding oneself funny and finding someone else funny. Trinitarianism would still leave you with a God that has no one else finding Him funny.)
I think the most obvious response would be the following reductio ad absurdem. Would having an infinite number of persons loving each other be a good property? If so, this argument would prove too much - God is therefore not a trinity but an infinite number of persons.
While we're at it, wouldn't having an external relation of love also be a good? So this argument also seems to prove there is someone external to God who is eternally co-existent and whom God (or the persons that make up God) loves. But it seems this is insufficient - it would be even better for there to be an infinite number of external persons God loves, so therefore there must be an infinite number of external persons co-existent with God whom God loves.
A more general response to these sorts of arguments is that highly abstract arguments like the one above (or the responses!) should always be met with skepticism, due to the ease with which human reasoning (including by the best human reasoners there are) can get things wrong. A survey of the history of philosophy will show how top reasoners come to very different conclusions on all sorts of things.
Following on 3., it is better to stick with scripture when trying to understand the nature of God. Jesus' words and example are explicitly designed to help us understand God. If we look at scripture, there is no clear teaching that God is a Trinity. The theology now known as 'Trinitarianism' took hundreds of years to develop, and hundreds more to work out problems that were generated by the Trinitarian theology itself. From the unitarian Christian perspective, it is not clear and evident in scripture, even retrospectively.