At the Last Supper, Christ took the cup and drank from it. Then He said, “Do this in rememberance of me”. At the Mass, the priest is acting in persona Christi, the person of Christ. He therefore must do those things Christ commanded, and one of them is drinking from the cup. If the priest has a cold, there are a couple options. He could only distribute Communion under one species. He could also, I believe, consecrate a small amount of wine in his chalice, and more wine in another chalice for an EMHC or deacon to distribute, and then consume all the Precious Blood in his own.