How do you eradicate the cold virus? It has never been done.
And won’t be eradicated now. It will settle down into an endemic rate in the population within a year or two and be just another of the multiple respiratory viruses we see every fall. A vaccine would be helpful but I suspect if we get one it will be like the flu vaccine. Helpful but not anywhere near 100%. It may even become an annual vaccine added to the flu vaccine depending on the mutation rate - that remains to be seen. Meanwhile there is no evidence all the economic self inflicted injury has done anything except harm many more people than will ever contract the illness. Even the most rabid proponents of lockdowns and social distancing admit it will not change the number of people who ultimately contract the disease - they just want to “flatten the curve”. something that they can’t prove the current charade is doing. Time to open up and get on with life.
If we wanted to eradicate "the" cold virus, we would go about it using many of the same techniques that we have used to eradicate other diseases. We eradicated smallpox through massive vaccine programs and contact tracing. We eradicated rinderpest (a measles-like disease that affects livestock) through vaccination and isolation of infected animals. We are on track to eradicate measles, polio, and guinea worm disease. For measles and polio, we are able to achieve this by vaccine programs and quarantines. There is no vaccine against worms, but we are on the brink of eradicating guinea worm disease through active surveillance and treatment programs.
It is not even necessary to have a vaccine to wipe out diseases. Every time there is an Ebola or Marburg outbreak in Africa, it is a brand new outbreak. That means it is not circulating in the human population. And we have successfully eradicated Ebola and Marburg every time there has been an outbreak.
I have confidence in the ability of public health professionals to eradicate Covid-19. Unfortunately, some members of the general public seem not so keen on eradication and instead have latched onto conspiracy theories and ridiculous notions of infectious disease transmission, causing them to work actively against any measures to control and eradicate Covid-19.
Oh, and the reasons we've never tried (and probably won't try) to eradicate the common cold is that 1) it is an annoyance but not fatal, and 2) there are over 100 viruses that cause common colds.