I agree that you are either considered a main Robin or you are not. I consider Steph a main Robin, and I think DC currently does as well. I hope I've shown that I understand that here tenure is different than Dick, Jason, Tim, and Damian's, but also why I think it's different from the We Are Robins, etc.
However, I would say that it is simply incorrect to say that Steph "can skip the role and she would be unchanged." War Games, for all its problems, is a defining story for Steph, and I firmly believe that guilt for how poorly editorial treated her was a huge part of why she was given the Batgirl role for two years, despite other characters (Babs, Misfit, a few others) being seriously considered (see the Editor's Note included on this page:
https://stephaniebrown.fandom.com/wiki/Batgirl_Vol_3_2). Without the prominence and sense of deliberately thwarted potential coming from her cut-short time as Robin, I think Steph would have stayed Spoiler when she was brought back in 2008. Instead, she became Batgirl, and reached a whole new audience.