I haven't observed that. It's a difficult position to hold: the ideas of karma and rebirth are central dogmas in Buddhism, and it's hard to construe them as not supernatural.
There is one major difference between Buddhism and some schools of thought in Hinduism like Yoga or Bhakti. While Yoga and Bhakti insist that a 'deity' helps you make the progress much faster on path of self realization, Buddha did not explicitly support this. He instead focused on more inward looking approach of meditation and contemplation (which is a subset of Vedanta philosophy in Hinduism). This is often seen as Buddhism being an atheistic philosophy by Hindus.
Both Hinduism and Buddhism allow its followers to embed this central concept into various other "decorations" and "facades". Which may involve supernatural elements. But they are mostly irrelevant for you to be either Hindu or Buddhist. This is in stark contrast to Islam, Judaism and Christianity where some central core beliefs absolutely require you to believe in supernatural elements.
This is nonsense. Buddhism regards the "self" as not really a thing; rather, a bunch of aggregates that beings mistake for a thing. The idea of a "supreme self" is Atman, which nearly all Buddhists regard as illusory. The word generally occurs in Buddhist literature in the form "anatta", or no-self. Clinging to the idea of a self is considered a mistake, and the root cause of dukkha, or suffering.
Similar are thoughtless (awareness) and mindfulness. Both may appear contradictory on the first glance but are same thing. These concepts are not easy to describe so often they are expressed multiple ways.
Rebirth largely depends on what kind of buddhism we are talking about. There is nothing supernatural about rebirth in Zen. Others not so much, Tibetan buddhism for instance they believe that the Dalai Lama is a direct reincarnation of the 13th Dalai Lama (who would be tried for pretty serious crimes if he was alive today). So meh. ymmv.
Treating karma as simply cause and effect amounts to an apologetic. The doctrine is that some kinds of action have effects that are borne in future lives. That would imply that there is some kind of place or medium where the seeds sown by actions can be stored, until they bear fruit in future lives. The Abidharma proposes some pretty convoluted metaphysics to explain this storage without postulating an enduring self.
Reincarnating lamas are a peculiarity of Tibetan Buddhism, honoured as much in the breach; for example, many lama lineages are passed from father to son. When searching for a reincarnated lama, monks rely on prophesies and magical signs. Unsurprisingly, the system is wide-open for corruption. Stephen Segal, for example, is supposed to be a reincarnated lama.