The purification theorem says that almost all mixed strategy Nash equilibria can be reconceptualized as close approximations of pure strategy Bayesian Nash equilibria, in which the relative probabilities of taking each action are nearly identical. Thus, if you don't believe that players don't mix in real life, this isn't a problem: mixed strategy Nash equilibria are essentially pure strategy equilibria in disguise.
Bonus note: John Harsanyi came up with the purification theorem. This was a major reason why he won a Nobel Prize in the same year of John Nash's victory.
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