"Hermann Bondi named the principle after Copernicus in the mid-20th century, although the principle itself dates back to the 16th-17th century paradigm shift away from the Ptolemaic system, which placed Earth at the center of the universe... Copernicus himself was mainly motivated by technical dissatisfaction with the earlier system and not by support for any mediocrity principle."
Copernicus' solar-system model is entirely separable from any assumptions about star size distribution, and the latter was apparently not a concern of his (nor do I recall seeing anything to the contrary elsewhere.)
On account of this separability, the mediocrity principle cannot be used to eliminate heliocentric models of the solar system from consideration, at least unless there's good evidence for it.
Nevertheless, the presumed huge size of the stars was seen as more or less of a problem (depending on which way one leaned on the heliocentricity issue), but it turned out that the apparent size of the stars was merely an artifact created by diffraction (the Airy disk) [1], making it possible to hold both that the sun is well within the range of stellar sizes and that other stars are far enough away that their parallax is difficult to observe.
Copernicus' solar-system model is entirely separable from any assumptions about star size distribution, and the latter was apparently not a concern of his (nor do I recall seeing anything to the contrary elsewhere.)
On account of this separability, the mediocrity principle cannot be used to eliminate heliocentric models of the solar system from consideration, at least unless there's good evidence for it.
Nevertheless, the presumed huge size of the stars was seen as more or less of a problem (depending on which way one leaned on the heliocentricity issue), but it turned out that the apparent size of the stars was merely an artifact created by diffraction (the Airy disk) [1], making it possible to hold both that the sun is well within the range of stellar sizes and that other stars are far enough away that their parallax is difficult to observe.
[1] https://en.wikipedia.org/wiki/Airy_disk