Considering I have been to the Cleveland Clinic and the Mayo Clinic, I visualize a sprawling campus of top medical researchers. Unless someone prefaced it with something like “Minute Clinic”, where I might envision something else.
They were founded in 1921 and 1864 respectively. Look up the original definition of “clinic”:
1 : a class of medical instruction in which patients are examined and discussed
3a : a facility (as of a hospital) for diagnosis and treatment of outpatients
b : a group practice in which several physicians work cooperatively
Clearly aptly named for the time.
You have a point that smaller, cheaper “clinics” are more common now - but your incredulity that most people understand the prestige of the Mayo Clinic doesn’t affect the reality that most people do.