There is a lot of debate in Western Buddhism about whether rebirth is real or not. Sometimes it is said that rebirth is an Indian religious belief, and not necessary for practicing dharma. However I am not so sure about that. I think the meaning of 'rebirth' is that 'so long as we identify with those things that are subject to birth and death, then we too are subject to birth and death'. When seen this way, rebirth seems a lot less fantastic, less like participating in an endless series of Hollywood films that many people seem to understand 'rebirth' to mean. It is more that we are then subject to all the sufferings of creatures bound to the wheel of birth, decay and death, and driven by instincts to keep struggling for survival.
We can't simply run away from that, however. It is not as if we can simply step out of that, even at the time of our death, because the latent tendencies will then re-form another existence which is also bound to the wheel of re-birth. It is not voluntary, it is out of our conscious control. I think that is why Buddhism stresses 'mindfulness' which is to understand these deep drives which power the 'wheel of life and death'. But that understanding is not a simple matter, it is not like having a relaxing time or being 'free from stress' in the way that worldly people understand. If it were like that, then simply being materially well-off and not having any emotional problems would be the same as liberation. But the Buddha teaches that, even though we might be lucky enough to be free of stress now, even for a whole lifetime (although very few are), at the end of that we are still subject to change and decay, and so still bound to the wheel of samsara, and so whatever favourable circumstances we have now will one day be lost.
So I think understanding 'freedom from rebirth' is not actually a matter of whether you believe in reincarnation or not. It has a deeper meaning than that. It is about whether you are of this world, part of the whole cycle of birth-and-death, change-and-decay, rising-and-falling, that everything in nature is subject to. Nowadays we seem to think that 'natural' is good and wholesome, yet it is the case that everything in nature is subject to decay and death, even if it is temporarily beautiful, young and vital.
There is something that is beyond change and decay, that is not subject to the constant cycle of birth and death. It is something always new, never perishing. That is what the Buddha found and points to. Living in the light of that, realizing what that is and making oneself available to it, is the aim of the Buddhist teaching. And that is not something that is taught very much in 'Western Buddhism'. Many 'Western Buddhists' can talk expertly and at length about subtle and abstruse concepts and quote passages from all kinds of texts. But in the absence of the understanding of the meaning of rebirth, 'nirvana' means simply being happy in this life, not having anything to worry about, being relaxed. It doesn't really have a deeper meaning. And so their idea of Buddhism supports that condition very well. But I think it's because they don't understand the meaning of rebirth.
May all beings realize the truth of re-birth in this human realm.