How did old linguists in a pre medical screening world manage to figure out phonologies so perfectly?

Articulatory phonetics was indeed done before Palatography. And not just by the Ottomans: the Korean script Hangul originated in articulatory phonetics, and for that matter both the Sanskrit grammarians and the later Graeco-Roman grammarians had pretty much had it figured out.

And they could just as my students in first year were able to learn phonetics from me, by watching my mouth and thinking about their tongue positioning. Yes, we used diagrams like that too, but people do know what the roof of their mouth is, or their hard palate, or that bumpy thing just behind their teeth; they know when they are rounding their lips, and when their tongue moves to the front or back of their mouth. For the phonemically distinct places and manners of articulation of any language—just half a dozen each—you don’t need any more detail in location than what you can introspect by being aware of what your mouth muscles are doing.

Phonetic detail needs more than that. And phonetic detail is the domain of the palatograph and the spectrogram.