Although most Christian apparently don't have this attitude; I find personally it hard to reject-outright the idea of reincarnation.
This mainly because (it seems) that most people, through most of human history, have believed in the reality of one or another form of reincarnation - plus several of the more modern thinkers whom I most respect believe in reincarnation, apparently from directly intuited personal experience.
However, I find that the Gospels tell us that Jesus taught all Men are resurrected after death - not reincarnated - and make their choice of Heaven or Hell. On the other hand, the Gospel discussions of whether or not John the Baptist was some kind of reincarnation of a prophet seem to confirm that, at least until the advent of Jesus's ministry, reincarnation was regarded as possible - if not universal.
One way I make sense of this is that I think modern religions tend to fall into one of only two categories - either they believe in some version of reincarnation with spirits returning to inhabit a series of bodies; or else that each human spirit is formed at a time related to incarnation. But these two are not the only possibilities.
A further alternative is seldom known or considered - that an eternal pre-mortal human spirit was alive before incarnation, death and resurrection; in other words that the full potential span of human life falls into three stages: pre-mortal spirit, mortal incarnate, and resurrected incarnate.
(However, given the role for agency and choice, presumably it is possible to choose not to be incarnated, and to remain as a pre-mortal spirit. This would presumably be the situation of some angels - who are either awaiting incarnation, or else have - at least currently - declined the offer of incarnation. And it would be the situation of demons - who reject incarnation along with rejecting God's plan for creation and the Love necessary to its accomplishment.)
This three stage understanding of human life (which is the Mormon view) is the one I regard as true - and my interpretation of those modern people who believe in the reality of reincarnation is that they have not sufficiently seriously considered this alternative. That, for example, they have misinterpreted their intuited memories of pre-mortal spirit life (which may include historical actions in this world, and with people in the past) as being incarnated life. In other words, they remember previous spirit lives, but simply assume that these must necessarily have been incarnated lives.
On the other hand, since reincarnation was apparently a possibility for John the Baptist, it is also possible that some modern people happen also to be reincarnates who, like John the Baptist, are spirits that have returned to fulfil some particular function, do a particular job... So when such people seem to recall a previous incarnate life or lives, maybe they are correct.
I find it striking that so far as I know, all simple, tribal, hunter-gatherer type societies believe in reincarnation - in the form of a 'recycling' of spirits within the tribe over time. The concept is apparently that there are implicitly a fixed number of spirits (or souls) who are reborn some time after death - so that the same set of personalities recur across the generations. My presumption is that such societies self-understanding will have been broadly correct - so this would imply that there used-to-be a, probably universal, system of reincarnation.
Most sedentary (i.e. settled, non-nomadic) totemic and pagan societies apparently either believe in some version of reincarnation, or else they regard life after biological death as being something like Hades or Sheol; that is continued existence of the spirit or soul in a ghostly, demented half-life of present-awareness without agency. Again, I would tend to accept that these people correctly understood their situation - at least in essentials. So, it is possible that this 'underworld' represented the time in-between reincarnations; or that some people/s (e.g. the Ancient Hebrews or Greeks) chose Not to reincarnate - but remained in Sheol/ Hades... implicitly awaiting the Messiah/ Saviour.
If we accept that the situation up to the time of Jesus's incarnation (i.e. approximately the years BC) was as above - that biological death was followed either by a a kind of suspended animation like Sheol, or else a reincarnation from such a state. Then the further possibility is that this situation was changed by the work of Jesus; and from some point AD onwards - probably the time of Jesus's own resurrection - spirits were resurrected instead of being reborn.
This also applied to the spirits at that point in Sheol/ Hades - some of whom were resurrected at the same time as Jesus. But - given the importance of free choice - it may be that resurrection could be refused, and that some of these may have a job still to do as reincarnates.
If so, modern people who believe they recall earlier incarnations may either be recalling their pre-mortal spirit lives; or they may be people who recall an incarnation (or more than one) before Christ's work in making resurrection, and who have returned to incarnation for some particular purpose.