There are different factors for red appearance of different objects in the night sky. Two reasons which I know are
So do we depend on only the red appearance of galaxies to find distance?
Star red by nature
Example: red supergiants
They emit red light because of their surface temperature.
The redshift of a blackbody is indeed completely degenerate with its temperature. However, galaxies also have spectral features arising at set (rest-frame) frequencies. Typically these are spectral emission or absorption lines associated with the energy differences of particular atomic transitions. We can determine the redshift from these features.
However, many telescopes do not have spectrographs! Redshifts often have to be estimated just from galaxy colors (specifically, the ratios of light intensity received through a handful of different color filters). In this case, there is absolutely a possibility for erroneous redshift inference.
See also How can redshifted light be detected?
Object moving further away from us
Let's assume there is a galaxy which is close to us but still moving away from us very fast so even that appear red.
Such peculiar velocities give rise to redshift-space distortions. These can be sources of error if you are trying to fix the locations of individual, cosmologically nearby objects. However, their importance drops at high redshift, because peculiar velocities do not grow with distance (indeed they are smaller at earlier times).
Also, when studying the Universe at a statistical level, redshift-space distortions are more of a tool than a hindrance. These distortions are characteristically aligned along or transverse to our line of sight. If we assume that Earth is not at a privileged location within the universe, it thus becomes clear which statistical features are due to redshift-space distortions, and then they are useful because they give us information about the Universe's velocity structure.
