It still does. And it's easy too see once you understand the core precepts of Buddhism, that it's not based on rebirth or karma. Those two are just one possible way to see our existence in this universe.
Buddha started with the question: what is the cause of suffering? Desire. But desire is nigh unavoidable in life. Every member of society is likely to have many desires. Thus, everyone is likely to have pain, sooner or later. A person with pain is likely to propagate pain to another person, or more. This a major method through which suffering in society as a whole tends to grow. So we need a way to tell members of society to reduce the propagation of harm. Thus, ahimsa, i.e., do no harm.
But people aren't just gonna internalise such 'for the greater good' ideas, and especially not when they are in pain. Buddha recognised that people often need a strong faith system to root them to these ideas, instead of relying on pure rationality to prevail all the time.
Buddha accepts that we do not really know how the universe works, but this fact does not remove the need for a strong faith system, because humans are imperfect. So — and being a philosopher — Buddha says humans can believe in whatever faith system works for them, as long as the system they choose upholds the core precepts.
The cycle of rebirth and karma just happened to be the predominant faith system at his time, and it also happened to fit the requirements, so most of his followers — at the time — had an easy job of adapting.
But he also warned against believing such faith systems to be the truth, instructing that people must not blindly take his teachings as authority, and must instead examine everything through their own experience and reason through it. When questioned, what to do if such examination reveals a truth that goes against his teachings, he clarified that one must go with the truth. The authority of scripture was explicitly placed below that of one's own experience and reason.
Thus, until proven, the cycle of karma and rebirth is not necessarily true, not is it required to explain that life is suffering and himsa (harm unto) propagates suffering. It's just a convenient way to tie it all together, just as good as any other unproven faith system.
If you don't believe me that Buddhism works just as well without karma and rebirth, take a look at Zen Buddhism. Despite being a modern form of Buddhism, it still holds closely to core of ancient Buddhism. And Zen makes no claim for, or against, the concept of Rebirth. It basically goes, "Idk. Does it matter?". According to Zen, who you are in this life, ceases to exist when you die, and that is plainly observable, so why not live the best life while you have it?
As you can see, this faith system also works to uphold the core precepts of Buddha. And I don't think he'd disagree with Zen on this matter. In fact, he'd probably be glad that Zen has made awakening accessible to people who had no background in the Hindu ideas of karma and rebirth, and thus would have found it harder to reach a path to awakening if they had to adopt those ideas first.