I suppose that's consistent with the belief in the incorruptible bodies of saints, etc.
Exactly so. Actually, that's the whole point of all that. Even the remains of the saints that aren't incorrupt are venerated. That's precisely what a relic is. The assumption is that the remains--the physical matter itself--is sacred and will be part of the resurrection.
Even the "preserved" ones are clearly in no state to be re-inhabited without basically being rebuilt from scratch. That goes for the body of any saint you care to name where they still have the body.
This doesn't apply just to canonized saints, mind you. The belief is that everyone will be resurrected. This is why Christians, like many Jews before them, buried their dead in ancient times despite the prevalence of cremation among the Romans. They believed that the bodies would rise again in the resurrection. But not in a merely metaphorical sense. It really is the same body, the same matter. Were this not the case, then Christians would have no need to claim that Christ's tomb was empty. (Another concrete example would be Lazarus, who would have already begun to decompose.)
Perhaps rather than 'reincarnate,' you might think in terms of 'recompose.'
I would argue that it sounds a lot like the question of whether a body teleported with a Star Trek transporter is really still the same body or not.
In some ways, yeah. Even if the bones of the departed are raised and new flesh is created from surrounding matter, we run into a Ship of Theseus problem. I suppose once you believe in an omniscient, omnipotent God, believing that he can track down every atom that once belonged to a person is no stretch. But I'm not sure it would be necessary to take things so far. In terms of pure matter, on the molecular level, I'm hardly 'the same' as I was 20 years ago. Yet, I'm still the same person. Why should some admixture of different matter in resurrection be any different? I'm honestly not sure what the answer to this question would be.
One of the key texts for imagining this is Ezekiel 37. There the imagery of recomposing the dead body is very graphic. Of course, in the context the whole thing is a metaphor for Israel coming back to life, but Christians take it to point to the resurrection of all dead.
All in all, the exact mechanics of the resurrection are unclear. There are two essential elements to it, however: 1) The matter itself is renewed, made sacred even, by its connection to a sanctified humanity. The head of St. Luke in Prague, just to take one example, is itself sacred and maintains a connection to the saint. The renewal of humanity is really just the first step in the renewal of all creation. 2) The very same person will rise again with their real, physical body. The exact way that's to work is unclear. But it's precisely what is meant by 'resurrection' in the first place, which literally means 'getting up again.'