A lot of what you said is very accurate, although I'd couch that with the fact that a lot of people start SS and training programs, in general, a little fatter than they ought to be. The program and the dietary recommendations are pretty simple for SS, if you're emaciated, skinny, etc. then you need to gain weight. You have come to be this way, i.e. skinny, because your dietary habits and stimulus to grow have not been adequate to support a larger, stronger human. Therefore, you must eat, grow, and get strong. You don't have to get fat to do this, and I'd argue that if you do get fat you're doing it wrong. However, if along the way you do it's not really a big deal unless the fat gain messes with your head enough to make you stop doing the program. I don't see fat and strong being inclusive with one another...
On the flip side, if you're significantly overweight and starting training then the stimulus to grow big and strong is there but you do not require a very big influx of calories to allow this to happen. Instead, you can sustain strength gains for a good bit of time and recomp if you start to pay attention to your diet, which didn't happen previously.