I don’t think so.
The nasal discharge, the sneezing, the coughing, etc. is indeed how the respiratory tract tries to rid itself of environmental irritants like pollen, dust or smoke. For those kinds of substances, that’s an effective strategy.
But it’s not an effective strategy against cold viruses since they’re not all just lying on the surface of the mucous membranes where they can be washed away. By the time these symptoms arise, many are already replicating within your cells. In a relatively short time, your immune system will figure out how disarm this virus, and that is how you’ll get rid of the cold, not by expelling the virus.
So the virus has one mission before your leukocytes do it in: get out into another host. It does this by co-opting your histamine reaction to get spewed out into the world on your excretions.
If you take meds to limit your excretions, you aren’t going to be sicker for having more viruses inside you. You’re already massively infected. But you are foiling the virus’ master plan by keeping it from finding its way into another poor schmuck.