My assumption is that reincarnation can happen and has happened - but that it is rare and exceptional - and very far from the normal thing for most people.
I state as a starting assumption that the Mormon view of life is correct - beginning as eternal primordial essences; we were made spirit children of God (that is, children of Heavenly Parents); who chose to be incarnated into mortal life and to die in order to enable spiritual progression towards deity - and that after our mortal death we are resurrected to continue spiritual progression, if we so choose and behave thus, towards the ultimate goal of full deity.
As an implication of this, there is a distinction between spirit, body and soul. We used to be spirit beings, but on becoming incarnated into bodies - en route to resurrection to eternal beings with bodies - the soul became a combination of the spirit and the body.
This is why mortal death is a terrible thing from which we needed to be saved by the work of Jesus Christ - because after incarnation, the death of the body is a tearing apart of that which has been fused, and it leaves the remaining spirit in a maimed state. This maimed state was described in the inhabitants of Hades or Sheol along the lines of being a demented ghost, without self-awareness or agency.
Resurrection is therefore a restoration of the unity of body and spirit which we have in mortal life - but (if we accept this gift of Christ) in a perfected and purged unity suitable for eternity.
In some primary sense, resurrection is a restoration of that 'old body' which the spirit was fused with in mortal life. So that resurrection is to be reborn with the same body - but that body made divine and immortal.
My point is that therefore reincarnation can only be either one of two:
1. Restoration. A return of the resurrected (purged and perfected) person to the world of mortal life - with the 'old body'.
The resurrected soul is thus restored to the mortal world instead of going to Heaven - presumably to do some divinely-appointed task (eg. be a prophet or angel).
2. Recycling. Some situation in which the first 'attempt' at mortal life is simply 'scrapped' - and the now-maimed human spirit is resurrected (i.e. has the integrity of its incarnated wholeness restored) but with the 'old body' restored but in a non-divine form.
With 'recycling'; presumably the first 'attempt' at mortal life had been such a failure (for whatever reason - perhaps the person did not live long enough to achieve what was necessary) - and they are simply given a second chance (if they choose to accept it) to experience mortal life a second time, and with what is in some essential fashion the same mortal body they had the first time.
My point is that if reincarnation is to be a return of a person to the mortal world and with a body, then that body must be the old-body restored; either restored in divine form or restored in a non-divine form - but the same body. The fact of having-been incarnated means that it can only be the same person if he also has the same body; and anything other than a completion of the same body represents a maiming.
In other words, our premortal spirit is not incarnated into mortal life as a ghost in a body-machine (the ghost being the real me, and my body merely its temporary dwelling); but that incarnation works by a profound process in which the spirit and body are made-one so intimately and wholly that they can never again be severed without loss of identity.
This is why true eternal life of the human soul is, and can only be, in the form of resurrected beings with bodies. But the consequence of this is that reincarnation can only be a very limited process, to achieve limited goals, as described above.
What is not possible, I think; is that you or I could be reborn multiple times, in multiple bodies, in order to experience multiple lives for the purpose of some kind of stepwise spiritual progression.
This is impossible if my assumptions are taken as correct; because such a view of repeated reincarnation necessarily assumes that the soul is actually a spirit, which can - and does - inhabit multiple bodies like a diver putting-on and taking-off multiple diving suits.
The classic view of reincarnation is therefore one which requires that we are ultimately, in our essence, only-spirits - and that bodies are no more than secondary and exchangeable garments. And this seriously challenges why - if we are spirits - eternal divine life should be as resurrected and incarnate beings, instead of simply remaining as spirits.
Indeed, the reason for mortal life becomes hard to understand at all. Why should we want or need to gain experience as mortal incarnates (with bodies) if our eternal essence does not actually need a body? Why be resurrected, indeed why be incarnated in the first place?
So - my belief is that resurrection can occur, and has occurred - but only seldom and as an exception. The normal and destined progression is that each life goes through incarnate mortality only once, en route to incarnate immortality and aimed at increasing fullness of deity.
This further implies that the full benefit of mortal life can be, and usually is, achieved from a single mortal life and death; therefore, given that most humans in history have died in the womb or as young chidlren, the main and only essential benefit of mortal life must be the experience of death.