At about the same time that the Upanishads were being composed and driven by very much the same ``protestant'' movement that was rejecting the largely ritualistic pantheistic Vedic Hinduism of the fifth century BCE in favor of a more spiritual and intellectual search for a pandeistic vision of Universal God, a back-country prince grew disgusted with his life of wealth and pleasure in a world filled with suffering. Unable to rationalize an infinite cycle of rebirth into a world filled with suffering, he gave his crown back to his father, left his wife and children, and took off on a quest for Enlightenment that was all of the rage at the time. The countryside was full of itinerant preachers, each with their own body of disciples, each claiming to have more or less of the ``truth'' about All Things. They were supported by alms, engaged in all sorts of ascetic practices that were supposed to lead to realization of the Truth (always the capital T, of course), and would alternate preaching in exchange for food with vigorous debates with gurus of the various competing schools that also claimed to have it.
As the legend has it, after forty years of mortification and discipline and meditation and effort, this prince-turned-disciple one day just plain gave up. He'd starved himself, taught himself to ignore heat and cold, sat in approved positions for hours, recited what he was taught to recite, and come up empty. Nothing he had experienced felt like Enlightenment, and nothing at all explained how to deal with the dilemma that had sent him to the road in the first place - the existence of suffering. In spite of following many rituals of many Gods, in spite of his trying very hard to realize his Atman as Brahman, outside of a literal appreciation of the words he had accomplished nothing like a direct knowledge of their reality.
He left his tree, went and bought himself a big meal, and satiated, lay down for the best sleep he'd had in forty years. When he woke up, he suddenly realized that food and sleep had done what forty years of effort had not - led him to direct, transcendental Enlightenment. He had long since discovered and enumerated the many causes of suffering - suddenly he found the solution to it. Suffering, he saw, was strictly in the mind. He had long since learned to discipline his mind - his ascetism had taught him that if nothing else. By thinking the right thoughts, engaging in the right actions, living the right life, one could avoid suffering. By acting on the causes, one could eliminate the effects!
He immediately went out and began (somewhat reluctantly) to teach his great insight, and found that nearly all of the ascetics he encountered on the road were similarly disillusioned; when he presented his rational plan for self-discipline plus compassion to eliminate suffering, they were all over it. Many were ``Enlightened'' on the spot, and became his disciples and later spreaders of the good word themselves. Almost against his will, he was forced to organize simply to accomodate the influx of seekers who were attracted to his words. Alone of all his competitors before or since, he taught his message on the pure basis of empiricism. Do not believe me, he said, just because it is me that is saying it. My authority means nothing. Try it out, and if it works for you then believe that what I tell you is true.
Buddha lived an entirely natural life, and we can be almost certain that he worked no supernatural miracles (although his existence has inevitably been mythicized, a thing that I think he would have found most distasteful as he was a man pursuing truth without the capital that means ``theistic scripture'' or ``bullshit myth''). He'd simply made a wonderful discovery in human psychology and ethics and wished to share it. This discovery is also noteworth as being an entirely atheistic discovery - it was very early empirical science, and neither God nor Gods nor human souls played any significant role in it. The prevailing religious structure, however tolerant it generally was of heresy as there wasn't any particularly strong system of orthodoxy in place to combat it, eventually grew weary of this competition that said that worshipping the Gods with appropriate rituals and gifts to the priests wasn't necessary, that the implicit serial rebirth with cosmic justice wrought upon sinners was unlikely, and that the priesthood itself was pretty much a bunch of Unenlightened exploiters that needed to work on suffering themselves. They (apocryphally, at least) arranged for Buddha to be poisoned as he stayed in a small inn far from his supporters, and that was that.
Buddha made pretty much no assertions about God at all, except to warn his followers that worshipping Gods and arguing about Gods was a waste of time and to be avoided. He denied the existence of the soul, and was lukewarm to the idea of serial rebirth, denying that he at least could remember anything from previous lives and so what did it matter if he had had them? He had certainly not lived a blameless life - his wife never forgave him or became one of his ``converts'' for walking out on her and abandoning his responsibilities in an act of supreme narcissism as if his Enlightenment were important, while she still had to deal with changing diapers, cooking, living very much for others (and probably living a far richer and more satisfying life than the Buddha thereby, I suspect). He had a lifelong ``problem'' with women and only established a path for women in his philosophy after one of his followers talked him into seeing the necessity of it - otherwise he thought that they were a pointless distraction of the men (who were important) as they sought Enlightenment while following his way. Finally, as great as his discovery of empirical psychosociology was, he failed to transcend his discovery and invent an actual scientific method, or to apply his insights to other aspects of the world.
After his death, his sayings and teachings, his most famous sermons, were preserved for many years within the memories of his most committed followers. As this body of first-hand memories began to drift and decay from the inevitable process of death and information degradation in the transmission chain, his surviving disciples (supposedly) all got together, pooled their memories, and built up the most accurate compendium of his actual words and actions that their memories permitted, using a fairly strict criterion for what they included. These were codified and rituals were established to ensure their precise retransmission, rituals that worked well enough that there is considerable correspondance along distinct transmission chains, suggesting that the entropy content is still fairly low. The best known collection of these sermons and sayings are the Pali Canons, although in the 2500 years since there have been philosophical splits and distinct schools built up around the mythicized Buddha, schools that have greatly exaggerated the cosmic significance of the whole thing and that (by their nature) have opposed any sort of mutation or natural evolution of the core sayings and beliefs.
At the same time, the local Hindu establishment immediately co-opted Buddha as an avatar of Mahavishnu who ushered in a new Yuga, thereby making him their own, end of story, case closed. Many converts could not get this idea completely out of their heads; Buddha must have been somehow special or else why is everybody following him? Buddha is therefore revered as something between a philosopher, the world's first clinical psychologist, and a god, even within Buddhism itself.
The latter, at least, would have shocked him and directly contradicts his teachings. The mere thought that people called ``Buddhists'' would sit around gilded statues representing himself and burn incense ``to'' Buddha while chanting rote statements in a dead language that most of the chanters cannot even understand would have appalled him, I think. Buddha would never have been a Buddhist, as the Enlightenment he preached doesn't come from ritually emulating the Enlightened, it comes from practicing mindfulness and living compassionately and without attachment, which doesn't require incense or chanting at all.
In any event, since Buddhism is not a religion, but rather an essentially atheistic human social philosophy or practice, since Buddha made no assertions of God or Gods other than to point out that if they existed as beings bound to the wheel of time they were no more enlightened than you or I and hence were not worthy of our worship, obviously there is nothing inconsistent about Buddhism and this theorem. Indeed, Buddha probably had a direct intuition of this theorem (as have many others over the centuries), and recognized that the Universe itself was the only possible God, and that as such it did not matter what you did in your life as far as God was concerned. What mattered, and what matters now, is how you live your life in human terms: You can choose to live ``well'' and thereby avoid suffering or the infliction of suffering to the extent possible, or you can live ``badly'' and cause the suffering of others and unnecessarily suffer yourself.