Question

what is the main advantage of using R2 over error variance an index of how well our model predicts actual data?

Answer 1

We know that R squared is the fraction by which the variance of the errors is less than the variance of the dependent variable.

Whereas, Error variance is just the variance of the error terms in the regression.

So, when we use the error variance as an index of how good our fit is, we can not say what proportion of variation in the dependent variable is explained by the explanatory variable; but when we use R squared, we can find the proportion of both explanatory variable variation, and the error variation.

That is the main advantage of using R squared, over error variance.