The variance that is important in an ANOVA is not the variance of the response. The important variance is the variance of the error term.
If a significant treatment effect is in the model, then the variation of the response is much higher than the variance of the error term because the treatment explains some of the variation in the response.
The mean square error is the best estimate of the variance of the error term. The mean square error from a pilot study is typically useful when you want to perform power and sample size calculations for an ANOVA.