How can a methyl cation exist without 8 valence electrons
The way you put it that everything wants to achieve noble gas configuration is a good rule of thumb for predicting molecules that are stable and long-lived. However, not everything is in equilibrium. A methyl cation certainly exists, and it certainly wants to achieve the noble gas configuration, but it still takes time to do so. So we can conclude that the methyl cation is an unstable, short-lived, highly reactive molecule, which is true. And you are totally right to explain this with the tendency of having a closed electronic shell.