"As far as I know, no studies have shown that evidence-based medicine leads to better patient outcomes"
Actually, every medical study shows that using the new method leads to better patient outcome than the old method (or that the proposed new method leads to worse outcomes).

"[no studies have shown that] companies which practice comprehensive A/B testing are more profitable than those which follow their intuition."
The question an A/B test answers is which option A or B will be most beneficial, which companies then use as a tool in their decision making. Companies are not stupid, if they think the resources required to answer the question with an A/B test are more then the benefits from knowing the answer, they would not invest in asking. I also suspect that the reason A/B testing is so common is that companies found that using it greatly decreased the risk associated with trying something different.

"The evidence that science is responsible for stuff like increased life expectancy is surprisingly weak."
I do not have access to the linked book, but it is my understanding that increased life expectancy is a result of improved hygiene, and a year round food supply. The food supply can either be got with science or a smaller population. Native Americans had excellent hygiene, and as a result relatively long lives. Europeans, in contrast, thought that bathing was unhealthy, so by modern standards they had poor hygiene. Without science, how can we determine who is right?

"This [cooking eggs in copper boals] was not something that chefs had ever established as scientific knowledge"
No, they established that through generations of cooking eggs in what was handy, eventually realize that eggs cooked in copper tended to be better. At this point, a couple of curios chefs might have cooked a batch in copper, and an identical patch in silver, and see if the copper ones really were different. That sounds like science. (In fairness, an explicit experiment only speeds up the prosses, at the very least you have evolutionary experimentation where those chefs who cook eggs in copper tend to be more successful)

" If chefs were forced to follow 'evidence-based cooking', not using anything special like a copper bowl until their was a peer-reviewed double-blind randomized controlled trial proving its effectiveness, the result surely would be worse food."
Finally, half a sentence I agree with. But before that, professional chefs do follow evidence based cooking. When they have a new idea, they cook it up, taste it, improve it, get a final draft. They then make sure there recipe reliably produces the same results, and gets several of their peers (or employers) to taste it and confirm that it is good enough to get on the menu. Even in the early stages, the decisions of how to make it were informed by past experience of what works and what doesn't, and I suspect there was a great deal of experimentation to get to the final product. The standard of double-blind trials is overkill in this case, because it slows down progress in exchange for making sure you do not go backwards. In medicine this is good, because we don't want to risk killing someone, however in cooking the risk of serving a bad dish is small enough that you can serve a new dish with less evidence it would be good.

"Can Steve Jobs provide a proof for the rightness of every iPhone feature?"
No, but I suspect Apple performs usability testing.