For sure, both immersion and deliberate practice work -- eventually. The issue is the scale and scope of the remedial project.
OP casually shunts off the severely deficient students as an 'invalid concern' and "just [a] transfer of responsibility", as if the schools merely "should teach them". Or, perhaps add an exam that would be solved by some 3rd party exam-cram courses or YouTube videos.
OP is wrong in this by orders of magnitude.
The training that you propose with immersion and deliberate practice is more, but likely still drastically inadequate.
There's little doubt that the surgery professors are observing a new and serious deficiency, even though the article doesn't mention quantifying the observations or research that nails down the exact cause.
Moreover, the skill deficit is deceptively simple, and may require years to fix, or even be unfixable in those subjects. Several examples:
* A group of monkeys being raised in an indoor arboretum had some injuries, and the scientists installed much more stout sticks and secured the vegetation. No more injuries due to broken branches, but when the monkeys were introduced to the wild, instead of swinging & leaping through the leafy canopies like their wild brothers, they cowered at the trunks. They'd completely failed to learn while young the skills required, and so the release failed for that generation.
* Personal experience, in learning a physical skill to an internationally competitive level, starting from being already well-coordinated, it took close to a decade of full immersion & dedication to get to the national team level. Moreover, even having achieved this skill level in certain sports, I've tried to get good at things like juggling and piano, and I have no illusions that even making a similar level of dedication, I'd never get anywhere near a professional level in either -- I didn't build that kind of wiring earlier.
* I've made arguments like yours to concert-level musicians, i.e., that simply being dedicated enough and training hard enough would get one to a professional level. Instant response of "NOoooo..." -- they've each seen multiple people pour years of their lives into hard work with top teachers and never get to a professional level, and these weren't people starting with a deficit.
* In neuroscience there is observed a period of plasticity for some skills. While it is not absolute, it is phase in brain development where the brain is particularly suited to absorb & learn certain skills. With some skills and animals, it is absolute -- if not exposed during that time, the organism or person will have an unsurmountable lifetime deficit. Languages are like this, where picking up a language between the ages of 4-8yo is almost effortless, but after that... Even if it is a skill that can be overcome in a different period of life, it's much more difficult.
Serious professional level manual dexterity and physical skills are not something that can be picked up even with multiple dedicated courses.
I'd venture that putting in the courses of immersion & deliberate training you suggest, and at a level of intensity equal to med school itself, might recover 30% of the deficient students in several years, delaying their Med School by the same period. Seems out of scope to me.
But the OP's idea of a pre-screening exam is good, not because prep courses would solve it, but because the pre-med students could wash themselves out of contention if they lacked the skills to begin with.