I don't think it would have worked if Ashoka wasn't Anakin's Padawan, not only was the Master-Padawan relationship important in developing both their characters, but it was important that they mark her as a main character.
Also I'm pretty sure Padawan's at this point are only allowed one master.
I also don't feel there is any room for her being mentioned in ROTS... at least in the scenes depicted.
And as the OP said she quickly became one of the strongest and most important aspects of the show, so I definitely think it was a good story choice in the long run.
Well there is no room for her because before the show came out there was no her. thats the problem. It reeks of pandering to me. I dont honestly know how it could have been done different, I am sure other stories could have been told with out her, but whether tehy would have been good or bad is anyones guess.
Like I said, dont hate her, just the concept of her. i dont think she should have ever existed, but now that she does at least i can use her to explain the other one that I dont think should exist as a legend.