Stannis burning his daughter was, IMO, their crowning fuk up. That was the most out of character thing they could have done with him (I could seriously see Stannis bending the knee easier than I could see him burning Shireen), and the show writers always had it out for his character so that scene always felt like spite as opposed to good writing. This is a guy who went to the ends of the earth to save his daughter from Greyscale just a few short years prior. I also remember a scene where her mother was going to smack her but Stannis stopped her and told her no one would lay a hand on his daughter. His love for Shireen was definitely integrated into his character in the show, and there was nothing believable about him burning her. I could see him burning himself while leaving others in charge to put Shireen on the Throne. I could see him forcing his army to continue marching through the snow until they either die or reach Winterfell. I cannot see him burning his only successor (the future rightful Queen) who also happens to be the only person in the world he truly loves.

At least he won't do it in the book. Book Stannis purposefully left her behind to keep her safe from his war. He also has both the Greyjoy siblings imprisoned so Shireens kings blood is unnecessary. There is seriously no legitimate reason he'd do it.

Stannis burning Shireen is where he diverged from his book self for the first time, and it was drastic. BookStannis also signed and sent a contract to the Iron Bank of Braavos to purchase 20,000 sell swords pledged to avenge him and put his daughter on the Iron Throne if he dies in the battle for Winterfell. He left her behind at Castle Black as a backup plan to fulfill their families duty should he fail and die. ShowStannis, on the other hand, made a decision to burn her in a single scene for a sunny day. It doesn't feel consistent with the kind of choice he'd make at all in the face of a single small setback (oh no its cold and my horses are dying).

Book Stannis when confronted with freezing to death: "Half of my army is made up of non-believers. There will be no burnings. Pray harder"
Show Stannis when confronted with freezing to death: "Let me alienate the fuk out of my entire army by burning the love of my life to death for a glimmer of a chance at something that might not even work"

Also, not relevant, but I think Stannis is going to beat the Boltons in the book. He has a history of overcoming the odds and beating his enemies in their own element and the book doesn't portray the struggle as one sided. The Boltons are also going through all kinds of chit and betrayals. Stannis has supporters on the inside. The battle also doesn't take place during a sunny day. GRRM claims the book is going to open up with what he calls the Battle in the Ice, so Stannis is going to go IN in a full out winter battle