I am not sure how many people in the modern world really believe in reincarnation; because so much reincarnation talk seems to operate at a superficial and 'lifestyle' level... Something to chat-about and speculate-on - or a stick with-which to beat mainstream Christians.
But presumably some people at least have reincarnation as a deeply-motivating kind of belief, that might sustain courage in the face of adversity?
But I must admit that I find it hard to imagine why a Christian who believed in Heaven (at least, who believed in heaven as I understand it to be) could want to be reincarnated after their biological-death, instead of being resurrected.
To my way of thinking, reincarnation is a natural and spontaneous way of thinking in childhood and during human history - and therefore I suppose it to be true: I suppose that Men (or at least some Men) were reincarnated after death, through much of human history. Reincarnation is therefore true, or a real possibility - or, at least, it was.
Although I also note that beyond the mere fact of reincarnation there are many and very different 'schemes' of reincarnation. Perhaps there were different reincarnations in different types of human society? I tend to think this is likely.
But what I do find difficult to understand about reincarnation for a modern Man (although here I will make an attempt to understand it) is why someone who knew of the reality of Heaven and the possibility of his own resurrection - and who also desired resurrection into Heaven...
...Why such a Christian would instead want to defer resurrection, and be reincarnated, and live another life in this world (in which this current life would not be remembered)?
When Heaven is both within one's grasp, and is wanted as an ultimate destination (and a situation in which the real business of living can begin, full-time) - it seems like a strange choice to defer entry.
I know-of, and greatly esteem (overall), several real Christians who also believe-in reincarnation, and apparently want to be reincarnated - who believe in reincarnation as both true and good: examples include Rudolf Steiner, Owen Barfield, and William Arkle - who are among my spiritual mentors.
This is find it hard to understand - because at best it seems like merely delaying - putting-off an achievable perfect outcome available Now - in order to engage in yet-more preparatory stuff.
But at worst it risks that my next incarnation might choose damnation and reject Heaven altogether - which would be the ultimate disaster.
However it may have been in the past; the hope of reincarnation nowadays strikes me as akin to kicking-the-can; as if just wanting to delay and defer the unavoidable and final decision.
And that strikes me as rather uncomfortably close to that delayed repentance, that refusal to repent Now; which is actually just a disguised refusal to repent.
The plea of Augustine of Hippo "Lord, make me chaste - but not yet" has often been misunderstood as a viable life-option for Christians. Of course, it merely means that Augustine was not yet a Christian when he said that (and meant it).
Analogously, when thought-through to its implications; for a modern Christian to desire reincarnation after death seems close to asking God for "Salvation - but not yet!" - which may well be functionally identical with rejecting salvation.
Note: It may be that some Christians regard reincarnation as something that just happens, that God 'does to us' (for our own good) whether we want and choose it, or not. Something that we need in order eventually to be allowed into Heaven and to assume the place God desires for us. If so, then this would surely be a cause for sadness and an attitude of resignation to God's will? Yet, many of those who argue for reincarnation clearly do not see it as a sad thing thus to be compelled to delay our admission to Heaven - on the contrary, they apparently have a positive and enthusiastic interest in the subject. This seems to me to display an implicit positive preference for reincarnation as their personal destination post-mortem - which I what I am criticizing here.