Even then, there are plenty of people whose MCAT scores were high enough to get them into medical school who ended up being terrible doctors, just as there are plenty of people who barely got into med school who ended up being terrific doctors. I can't cite any studies, but I imagine that just like anything else the variability within a score band is just as much as the variability between score bands.
(Similarly: plenty of bad lawyers have passed the bar, just as plenty of good lawyers failed it before they passed. There's enough variability between the Bar exams of different states to make this possible. I suspect the same is true about engineering programs as well).
It may just be that the MCAT is a good indicator of one's ability to become a doctor. Med school admissions are so competitive right now that these schools are able to be pretty selective about who they admit.
It may be that these tests were designed at least in part by the professionals in those respective fields to measure knowledge and competencies that directly relate to the domains in question.
More to the point, this is a specious comparison because the exams you cite are professional vocational exams which correspond to a specific career path/domain.
There's no such vocational test that will adequately cover every child, nor do we even have philosophical agreement about the skills necessary to live a good life, much less the relative importance of such skills, and even if we did, there are so many different ways that things like literacy and numeracy manifest that it'd be downright impossible to construct a test that can even come close to measuring it.
The tests you cite are tests that limit who can practice in a given profession, and while they're not perfect, I'll concede their usefulness. More importantly, they're on an individual basis. There'll definitely be some false-positives (people who are good at taking tests but poor at performing the basic skills necessary for competency) and false-negatives, but those can be corrected.
It's a whole different ball-game when you're talking about a non-vocational test, particularly one aimed at people who are not yet adults and who have not yet chosen a particular career path.