This is going to be an answer with less thorough sourcing, as I can't find the discussion of the common cold syndrome I recall in either Harrison's or Cecil, which is a shame, because, as far as I can remember, it was very good and really helped me get a grasp on how to approach clinical syndromes in infectious disease.
The common cold as a clinical syndrome is not any particular viral infection, but a cluster of symptoms that follow a stereotypical course. It's generally associated with an initial viral infection and you can isolate infectious particles early in the course of the illness, but by the time most people go to the doctor there is no active infection. Because of this, care for a common cold is directed at the symptoms. You could even say (and I do) that the common cold is the symptoms, not the infection.
The development and resolution of many common cold symptoms are unrelated to direct infection. Cough, for example, when associated with the common cold, is not due to infection directly, but reactive inflammation, and is one of the symptoms that lingers the longest. Treatment limiting postnasal drip and, in patients with reactive airway disease, any associated asthma flare up can shorten the length of the illness.
So, no, treating the symptoms in the case of the common cold is not a bad idea, and does not cause the disease to last longer. It may even shorten the disease course.
You can see some of these principles discussed in these clinical practice guidelines and in this review of the pathogenesis of one of the viruses most often associated with the common cold syndrome.