Is eradication of the Common Cold possible?
You'd think there would be numbers of papers on this. I have two references but they're not great.  The answer is not really.  Many people carry cold viruses asymptomatically.  This can be either a natural immunity, or that their body has beaten it down but not entirely eliminated it.  Cold viruses do not survive long outside of the body as you stated.  The only way they've made it this far is some survive inside the body until they are spread to other people.  In essence we are the reservoir.  So you could isolate everyone for two weeks, but that won't cause a problem for the viruses that are just tagging along.  There may be other reservoir animals as well for many of the viruses.  But even those that only affect and are carried by humans would survive.

The group of cold viruses is just too varied and different to eliminate them.  Also the cost of such an effort would not really be justifiable given the typically mild symptoms.

Natural Reservoir