It seems that reincarnation goes-with a belief in the superiority of spirit - the superiority of existence as a spirit over incarnated existence...
I say 'goes-with' because I don't think reincarnation logically-implies the superiority of spirit, but goes with it in a natural kind of fashion - apparently; if such versions of reincarnation are considered as Hinduism, Buddhism and recent doctrines such as Anthroposophy and some New Age ideas. By the 'superiority' of spirit, I mean that with reincarnation it is usual to see life as a pure spirit as superior to life 'in' a body: the body is seen as a restriction.
(For example, Rudolf Steiner envisaged a long-term evolutionary process by which Man would become incarnated, multiple times - with a more and more 'solid' incarnation; and then the process would be reversed - but with enhanced consciousness - until Man was again fully spiritual. A U-shaped evolutionary descent into, and out-from, matter over many thousands of years. I am not sure whether this was also Owen Barfield's mature theological position - but I think it was.)
For reincarnation, repeated incarnations serve the life as a spirit - and usually the ultimate goal is to stop reincarnating, discard bodies, and live permanently (finally, eternally) as a spirit.
The incarnations can serve spirit in various ways - each reincarnation might provide an experience to allow spiritual progress, or be a kind of opposite of this - the incarnation being a punishment or adverse consequence of earlier lives... but in the end the idea is that these incarnations, these repeated embodied lives, are merely a means-to-an-end; they have the purpose of ultimately allowing the body to be discarded.
In Christianity the picture is different - but there is here a difference between 'Mainstream' Christianity and Mormon Christianity.
In most kinds of Catholic and Protestant Mainstream Christianity, the Trinity of Father, Son and Holy Ghost include two spirits (Father and Ghost) and one resurrected incarnate (Son). The overall sense seems to be that whether God is a spirit or incarnated makes no difference - since there is (by the mystery of the Trinity) a unity of all three.
But there is a implicit sense in Mainstream Christianity that being incarnated is a problem. Man is regarded as primarily incarnated, as beginning as an incarnate - and the problem of incarnation is solved by means of Jesus descending into the incarnated state, dying and being resurrected... The feeling is that incarnation is a problem that needed solving, and was solved by such means.
For Mormon Christianity, incarnation is superior to spirit life. The Father is incarnate - God is not 'a spirit' but has a body.
Men have their (pre-mortal) origins as spirits, and incarnation is seen as a necessary step in progression to full divinity, to become like the Father.
(Not all pre-mortal spirit Men have incarnated, and presumably not all will necessarily incarnate - if they chose not to. They would remain as spirits - as angels; or as demons, none of whom are - according to doctrine - permitted to incarnate. Therefore, for Mormonism, incarnation is a privilege.)
And Christ too (although highly-divine as a pre-mortal spirit) necessarily went-through this stepwise process in order to become fully divine, like his Father. But instead of dying and becoming a spirit (as happened to all men before Christ) - by dying and resurrecting; Christ began the new era in which all mortal incarnated Men died and were resurrected.
For Mormonism, incarnation is superior to being a spirit (all else being equal); in the sense of incarnation being a more divine form of being (and, as I said above, necessary for Man, including Christ, to become fully divine).
Therefore; for Christians in general, and Mormons in particular, there is no point in reincarnation - unless something has, in some way, 'gone wrong' with the primary incarnation (maybe that it was ineffective at achieving its purpose for some reason - perhaps extremely premature death?).
My conclusion is that - for Christians - reincarnation isn't a thing that is necessarily ruled-out nor false... As I have previously noted, the discussion in the New Testament of whether John the Baptist was a reincarnated prophet - and if so which one - suggests that the possibility of reincarnation was acknowledged by Jesus and his followers.
It is more a matter that reincarnation is superfluous for Christians, but esepcially for Mormons - at least under 'normal circumstances. Only if something has gone wrong with the primary incarnation would there seem to be any compelling reason to have further incarnations.
This leaves-open the question of how often things go wrong in human mortal lives, such that further incarnations are required (or desirable). Is it common or rare? To that I have no answer.