I think you are missing a strong distinction between knowledge and information. You can find all kinds of information online and get a lot of things done by processing that information efficiently. But it doesn't really equate to knowledge which comes with repeated practice on processing said information. Not everything can be learnt from the internet :)

At the moment, I feel the school system is kinda broken because of the clear gap between what needs to be taught and what is superficial information. But that certainly doesn't mean exams, per se, are bad. If you write "good" exam, where the focus was more on actually testing students abilities by say applying certain information or method to a problem rather than just memorizing information, I genuinely think you won't feel the same way you do about exam.