Well, the OP was about a theorem, so I explained the theorem... Bayesian theorem formalizes the process through which human intuition works and with which we pick theories. It is hugely ironic that people not familiar with the Bayesian theorem often consider it unintuitive, since the theorem itself is the basic process of human intuition formalized.
Now, the whole point of the classic scientific method, developed before the scientific mainstream even knew about Bayesian theorem, is to "fight" human intuition since it (probably rightfully so) considers human intuition as something that is only useful to make a guess, but as not really useful (and one that often gets in the way) when one needs to confirm a theory. To summarize, the scientific method was developed to "fight" Bayesianism before Bayesianism existed. Hence people writing long articles and philosophizing about what should be a trivial issue.
But Baye's theorem is not obvious to humans. The intuition is P(A|B) = P(B|A). Base rate fallacy etc. etc.
That said, Bayesianism is not the same thing as Baye's Theorem. You can be a frequentist and still apply Bayes Theorems or the like. Bayes theorem is straight forward once you train your mind on it.
Bayesianism is much more than that and is not a trivial issue. But you are also correct in that human philosophical intuition on probability is Bayesian. With a very broken application. Few people that claim to be Bayesians are actually doing pure Bayesian probability.
Hand wringing over priors and model structure is what Bayesianism is all about. Philosophically, the viewpoints of bayesianism and how to best pick priors are interesting. Also interesting is how Quantum Mechanics fits neatly into the bayesian perspective.
I have a feeling that Bayes is unintuitive specifically because people pick extreme or otherwise bad examples to explain it. As you mention, the way we calculate odds is oddly (pardon the pun) similar to Bayes.
Here is an African Safari example. The probability of an individual hunter being attacked by a lion P(L|H) (assuming any lion who tracks down a hunter will also attack him) equals to the conditional probability of the hunter tracking down a lion P(H|L) (which is a measure of how good the hunter is at finding lions assuming there are lions) times the general probability of encountering a lion in the bush P(L) divided by the general probability of encountering another hunter like himself in the bush P(H). Hence why you can get away with hunting lions alone only if you're very bad at finding lions or if there are no lions.
Now, the whole point of the classic scientific method, developed before the scientific mainstream even knew about Bayesian theorem, is to "fight" human intuition since it (probably rightfully so) considers human intuition as something that is only useful to make a guess, but as not really useful (and one that often gets in the way) when one needs to confirm a theory. To summarize, the scientific method was developed to "fight" Bayesianism before Bayesianism existed. Hence people writing long articles and philosophizing about what should be a trivial issue.