Seems like you are implicitly describing something similar to the the backward stepwise selection method. Here you:
- first start with the full model with $p$ regressors and calculate the test-SSR;
- calculate the $p-1$ models with $p-1$ regressors, each time leaving a different regressor out, and select the model with the best test-SSR improvement;
- repeat 2) with the model previously chosen, updating $p \leftarrow p - 1$;
A different approach is the the forward stepwise selection method, where you do the same comparison between test-SSR improvement, but start with a single regressor and work up to $p$. The advantage here is that one can use it even if $n<p$, i.e. observations are fewer than regressors.
The two approaches are done when best subset selection is not computationally possible, i.e. calculating the test-SSR for all possible combinations. However, both backwards and forwards stagewise selection is not guaranteed to find the best model containing a subset of $p$.
Obviously, here we are using (cross-validated) test-SSR instead of significance. Other common measures are AIC, BIC, adjusted $R^2$. Doing this with significance seems to lead to a multiple testing problem, i.e. like a t-test done multiple times on different regressors instead a single F-test.
(I know it doesn't explicitly answer your question, I posted this as an alternative to your approach and maybe also to highlight some potential problems.)