1) There isn't a single virus called "the cold virus"
2) In addition to SARS-CoV-2, there are other viruses that cause cold-like symptoms that are not spread by fomites.
Your comment gave the impression that there's a single "cold virus" for which fomite transmission is the primary vector. That's not so. There are a bunch of viruses (in fact a bunch of families of viruses) bunch that cause cold symptoms, and only a few of them are commonly spread by fomites. None of the five endemic coronaviruses are known to be among these, nor is influenza.
Also, even though the CDC appears to be correct to my eyes in this case, I urge that, after the disastrous departure from anything resembling science in the past year and a half, we stop regarding it as a credible source.