To be really fair, native Eorzean medical knowledge appears to be somewhat rooted at a
medieval level, with a solid belief in long-abandoned archaic theories such as
body humours. Not to mention a lot of medical treatment is purely magical, using conjury and arcanima to heal wounds and disease (and by extension, white magic and scholary magic). Then there is alchemy for healing which is another branch of medicine which is perhaps closer to actual scientific principles of chemistry (as it was in the real world), but still with a esoteric bent.
As for the tech levels in Eorzea, Garlemald may have lots of techno gadgets but Eorzea itself is still rather primitive, with
uneven levels of technological development (much of which has been the result of Cid and the Garlond Ironworks gradually trickling in Garlean tech into Eorzean hands), with often even simple technological advancements such as a
bicycle actually regarded with suspiscion by Eorzeans, not helping change Garlemald's attitude towards Eorzea as a cultural and technological backwater inhabited by primitive 'savages' barely above stone age level.
Really, like everything in this game it requires a bit of suspension of disbelief to work, as there is a handy hint in the series title after all.

Don't think too much into it.