She actually did take over the Robin mantle during a time Tim couldn’t, but it was really short lived. If I remember correctly she was fired from being to impulsive for Batman’s liking. Afterwards Tim returned to the role as Robin. In terms of comic book history she is the first canon female Robin, even though her time was shorter than Jason’s.
I absolutely think she was Robin, and am really pleased to see DC building on that history with the Robin 80th anniversary special (though Firefly does agree with you there and calls her “cosplay girl”) and the Robins books (plus Robin #5 this summer).
From what I recall, she was officially Robin for a short time. I much prefer her as Spoiler or Batgirl, but that doesn’t change the fact that she was Batman’s sidekick. That partnership, even though brief, is a different relationship compared to other Gotham vigilante roles. I’m fine with Steph being included in Robin-centric stuff because she did experience “Robinhood,” which impacted her life. I’m also fine when she’s not included, if the story is brother focused or adopted kid focused - although the last includes other characters dependent on the era/timeline - which seems to be the aspect the OP is highlighting.