Factor investing can be explained by factor models, via the factors exposures. For example Fama-French observed that Size and Book-to-Ratio were systematic risks of a portfolio and consequently they led the following regression
$$R_i-R_f = \alpha_i + \beta_m(R_m-R_f)+\beta_s SMB + \beta_v HML + \varepsilon_i.$$
However, the thing that I don't get is how this equation allows them to prove that Size and Book-to-Ratio are systematic risks. Why did they regress with respect to SMB (size premium) and HML ? It would have been easier to just regress with respect to their size and book-to-ratio values directly, doesn't it ?
Why is it important to know the correlation between a portfolio return $R$ and a SMB/HML portfolio ? Are not we rather more interested in the correlation between a portfolio return $R$ and the size of this stocks instead ?