Stannis was never the underdog, his army would've crushed the Lannisters, even without the Tyrells, had it not been for the wildfire. After that he became the guy who lost and resented it and everyone (in my eyes anyway).
I saw him as the underdog because of the way he is introduced to us. He's isolated, discarded and unloved by those who, one might assume, would be the most loyal to him. Both of his brothers have disrespected him, but he remained loyal to them. When he needed that loyalty reciprocated, he's rewarded with Dragonstone from Robert and the usurping of his claim to the throne by Renly. Not to mention, he was all but ignored when he tried to gain support for his claim by traditional means. The only reasons he had the advantage going into Blackwater Bay were thanks to Daavos convincing Salador Saan and Mel pushing out two shadow babies!
In regards Stannis's percieved slights, I understand that much of this information comes secondhand from Daavos's perspective so it's likely to be intentionally biased in favor of Stannis. But come on...Stannis is a dick, but I can kind of understand why. Also, among the nobility presented in the book, he is the only one (that I remember) heeding council from people of "lower birth". He loves his disfigured daughter. He is outwardly conflicted about Mel's advice to burn Edric (hopefully I'm not conflating that with the show). He admits regret about his conflict with Renly...
The point is that Stannis, to me, is one of the best realized characters in the story. He can say things that can make the reader hate him (like being against Robb Stark), but later he will give some anecdote to help us sympathize with his frustrating cynicism. His character arc feels more like a character wave because he has had so many low and high moments in the book (and show) and I've enjoyed riding that wave.