Regression hypothesis testing: rationale behind using $\frac{SSR}{SSE}$? Goodness-of-fit testing in multiple linear regression based on a centered model usually uses the test statistics
$$\frac{SSR}{\sigma^2 p} \bigg/ \frac{SSE}{\sigma^2(n-p-1)} \sim F_{p, n-p-1}$$
, where $\sigma^2$ is the variance of the error term, $n$ is the number of observations, and $p$ the number of features.
I have two questions. First, what is the rationale behind using this ratio? For example, what if I only look at SSR or SSE, but not both?
Second, is there any other obvious choice? For example, what if we consider R-square of some variations of it? It's curious that no test involves SST. Is that because SSR and SSE are independent so it makes derivations easier?
 A: Part of it is the F-statistic is well-known.
I had a professor who pounded into our heads that any formulation/manipulation of the data was valid and met the criteria of being a 'statistic'.  But the challenge was using (or coming up) with ones with known distributions for confidence testing.
Because that statistic is distributed with known parameters as an F(-) distribution, it means you can readily rely on tables, papers, software, and theories. 
But nothing stops you from doing your own, provided the lead somewhere.  Imagine if you wanted to measure the height of the average Dutch person.  You could measure every person arriving from Holland on a randomly-chosen with a Dutch passport.  Or, you could measure the first person off the plane that meets the criteria.  Both are reasonable, and indeed unbiased, estimators.  But one has a lot more theory and practice around it.  
You are free to do as you choose in statistics.  But you must make it work - and most people stick to proven formulae.
