What critical factor determines visibility for faint Deep Sky Objects (DSOs) versus merely making them appear larger?

Light-gathering power, dictated by aperture.

There is a fundamental distinction between viewing planetary targets and viewing Deep Sky Objects (DSOs) like distant galaxies or faint nebulae. While magnification makes objects look bigger, it does not inherently make faint objects visible; magnification simply enlarges whatever light has been gathered. The capability that allows faint, diffuse DSOs to be seen at all is the instrument's light-gathering power. This power is directly determined by the aperture—the diameter of the primary lens or mirror. A larger aperture collects significantly more photons over the exposure time, enabling the faint structure of a galaxy to resolve into a visible image, even if that image appears small.