It stems from a shift of meaning and emphasis
1620s, "bedridden person, one confined to his bed by sickness," from French clinique (17c.), from Latin clinicus "physician that visits patients in their beds," from Greek klinike (techne) "(practice) at the sickbed," from klinikos "of the bed," from kline "bed, couch, that on which one lies," from suffixed form of PIE root *klei- "to lean."
Also "one who defers baptism until the death-bed" (1660s). Sense of "private hospital" is from 1884, from German Klinik in this sense, itself from French clinique, via the notion of "bedside medical education, examination of a patient by an instructor in the presence of students."
The modern sense thus reverses the classical one, in which the "clinic" came to the patient. General sense of "conference for group instruction in something" is from 1919.
The idea of group instruction was extended from medicine to other areas:
a group meeting devoted to the analysis and solution of concrete problems or to the acquiring of specific skills or knowledge
The person offering or providing such a clinic would necessarily by knowledgeable and authoritative, so would become respected. Putting on a clinic became a symbol of such success.
The idiom putting on … occurs elsewhere as in putting on a show, putting on airs
to do an activity, esp. one that others can watch: