That's a nice fantastical narrative the Hindus and Brahmans have been pushing on morons, which the Jews and then Jesus continued to perpetuate, to try to come to grips with mortality.
Too bad it's bullshit.
Well it's not a required article of faith that you need in order for Christ to be saved.
Both incarnation and re-incarnation are phenomenon. By that i mean they are appearances that are perceived to the degree they are believed and experienced. The experience is evidence of your belief in them. Your belief in either phenomenon, or any phenomenon at all, must come to zero, in order for Christ to be saved.
You are right that belief in the phenomenon seems to have had some traction among Jews and early followers of Jesus, some thinking he was a re-incarnated prophet of some sort, for example. That view was changed by an official narrative from the Magisterium, that you only get one chance in a very limited window of time. This augmented the heaven-vs-hell narrative, ramping up the fear factor, all the more effective to modify behavior. So that is an example of priest-craft.
No, you don't need to believe in re-incarnation for Christ to be saved. You do need to disbelieve it as much as you do need to disbelieve in
any incarnation. It doesn't help Christiaints to believe in incarnation, and disbelieve in re-incarnation. Nor does it help any kind of atheist, theist or deist, to believe in incarnation, and to disbelieve in re-incarnation.
Believing in incarnation is enough to save the believer from Christ being saved instead. A believer in incarnation will experience the phenomenon of re-incarnation, whether they officially believe it, or not. At the highest level of the world, it
is believed. At the level of belief that turns the world in the first place, that is how the world continues to turn, and is not changeable at human level of faith. That is, faith has already established the way the world turns, and there is little the human-level believer can do except to experience it...or to completely reject every part of it. You can't accept some parts of the world, and reject others. Over time, you will experience it all. Salvation is about shortening that experience and bringing it to an end.
Technically, Christiaints are striving for re-incarnation, despite protests that it is not a phenomenon that can be experienced. Resurrection, to them, is when they get their body back. Re-incarnation is the net effect of that kind of belief. You do get your body back, but it just won't look like the old one, and you mostly won't remember the old one, and the new one just won't last as long as you hoped before it get's recycled again...and again....
Technically, bodily experience does last forever, until, or unless it is disbelieved. The only way to disbelieve it is to know what is true instead. It's not even enough to believe what is true...one must know it.
What is true is that there are no male or female in Christ. In other words, there is no bodily experience when one is experiencing the Truth as one's Self. The Self is Spirit, so-to-speak. Or some combination of Spirit/Mind. It is not balanced with a body. That is not "balance".
Since you do believe in incarnation (the flesh), you will experience all aspects of that phenomenon, given enough time. Re-appearing (in some kind of personality package) is part of that phenomenon. You're appearance in the flesh is the evidence that you do believe in incarnation.
Any religion which does not clearly explain the way to cease and desist re-appearances, and advocate for the full renunciation of the world, is not serving the salvation of Christ. Rather, it is serving the salvation of the flesh.
This is an example of renouncing the world: neither incarnation, nor re-incarnation are true. Both are equally a false perception based on self-deception. No part of the world is true, neither the beginning of time, nor the end of time. Planets are not true. Galaxies are false. Stars are liars. Stars do not emit real "light" nor impart any real knowledge. Humans do not have souls which can be saved. They have egos which can be perpetuated over time.
Only Christ has a soul worth saving. An ego is a somewhat invisible phenomenon which feels most at home in what i would call a personality package, which would be some kind of "balance" between spirit-mind-body. Body is what expresses an ego's ideology the best. This explains why as long as ego persists, it will always be endeavoring to reappear on a body of sorts. At the end of the day, even an ego is false, itself a product of faith. All phenomenon supported by faith will eventually disappear from one's experience. The ego must be renounced for Christ to be saved.