The bias is not really against Baratheons and its not on the side of Targaryens, it's character based. Daenerys may be favored more, but she is not the first character and she won't be the last. All Bioware games have favored characters too and we are in a Bioware forum right now. I think by now you can see some resemblance between Leliana and Daenerys.
I simply see more opposition to Stannis than I see to Daenerys. Stannis already has a pile of haters on Westeros but Dany will be new. Her family lands are in Baratheon hands, so no one will mind she getting them back. In fact Dragonstone has to be quite defenseless given the fact that Stannis has pulled vast majority of his men from there.
We didn't witness her story for all this time for nothing. I think her going back to Westeros and events that transpire after that is inevitable at this point, but not her getting the iron throne back. We'll see how the show handles Aegon. As stated he has a much better chance for success and people believing him if he and Dany act together.
Stannis, for the most part, is the only Baratheon, his daughter aside. Daenerys is the only Targaryen , even if I was talking about the Houses themselves, which I wasn't, there really isn't much of a difference. I'm not sure where the comparison to Leliana comes from, Leliana isn't a magical, inbred "godling" who hatches dragons and has merry adventures where she frees slaves and ends teh tyranneez.
Dragonstone isn't rightfully Dany's anymore than the Iron Throne is, so any opposition to that will also be opposition to her taking the heir-apparents seat.
As for the rest, even if Aegon is legitimate, which I'm 50/50 on, the line of Targaryen kings is just about done. I highly doubt either of them will take the throne in the end, just as I doubt Tommen will be King of the Seven Kingdoms.