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.
right here.... kind of.... I just didnt like the idea that Anakin had an apprentice since you would think something like that would have come up in the movies. It felt like pandering just trying to get in a new demographic rather then an actual good story choice...... and for the start thats exactly what it was.
heck you didnt even need to make her Anakins or Obi-wan's padawan she could have just been one of yoda's that didnt have a master and was just traveling from master to master or something and it could have worked.
So dont hate her, just the concept of her.
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.