I actually think math and sciences should introduce what I call "synthesis" much earlier. i.e. I don't think it's unfair to give students all the ingredients and add in a question on the exam to see if they can take those ingredients and apply it to a problem type they haven't seen before. (This is a great differentiator between C students and A students.) Or for a science class, rather than perform an experiment, I think the students should have to actually DESIGN the experiment first. (I had one laboratory exam in 2nd semester undergrad chem class that did this and it was amazing! The students also performed pretty well at it too. It consisted of being told to figure out how much zinc was in a lozenge. We were also maybe given a handy reaction formula and that was it. You had to design your analysis procedure and figure out how to get the quantity you wanted out of it, and then actually perform your analysis all within the exam period.)

I think not doing this starting in like middle school is a big part of the reason why people think math/science is useless. Unless the exact scenario they have been taught pops up, they can very rarely see the application. But the real world NEVER works this way. A problem is NEVER formulated as a straight forward well-formed problem. Figuring out how to mold it into something that you can apply the tools you know to is in and of itself a REALLY important skill to practice, and sadly, we almost NEVER practice that. Only in grad school does that type of thing come up.

Depends on your sense of fairness. Math Olympiads don't test what's in the book, but they are also fair.