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>"The school of Asclepius" actually folds together three different things that need disentangling:
>The Asclepieia — healing temples (Epidaurus, Kos, Pergamon, Tricca) — were religious sites. The sick came for incubation: sleeping in the sanctuary to receive a curative dream-visitation from the god. Temple healing, not rationalist instruction.
>The Asclepiads (Asklepiadai) were something else: family lineages of physicians claiming descent from Asclepius. Hippocrates was traditionally an Asclepiad of Kos. This is the closest ancient analogue to a guild in medicine — and notably it has exactly the features you're reaching for. Look at the Hippocratic Oath: you swear by Apollo and Asclepius, you treat your teacher as a parent and his sons as your brothers, you support him financially, and you transmit the art only to your teacher's sons and to pupils who have sworn the oath. That's an oath-bound, closed, quasi-hereditary brotherhood with controlled transmission and secrecy. So the guild-like structure does exist in medicine — but it lives in the Asclepiad lineage tradition, not in a Roman collegium.
>The Coan and Cnidian medical traditions then grew up associated with these centers (the medical school at Kos sat alongside its Asclepieion), so the temple cult and the rationalist medicine coexisted, sometimes physically adjacent, without being the same institution.

Kos -> Qaws -> Qos -> Quzah -> Yah
Coan -> Cohen
Asclepius -> David
Druids -> Asclepiads -> Qumran Essenes