Unless I've forgotten some line of dialogue, I never had the impression Stannis wanted to be king of his own ambition - rather, once he learned he was the heir, it was his duty to be king, his right, and that right drove his actions, along with a certain redhead fanning his flames. To me, his ambition grew out of that right. If Joffrey had been a true Baratheon, I don't think Stannis would have ever risen up to challenge his nephew's claim, unlike Renly usurping Stannis's claim. I don't think Stannis ever wanted to be king instead of Robert, though he did resent that his younger brother became Lord of Storm's End rather than himself. Regardless, I think he viewed the battle in which he died as his duty, but also as part of his overall ambition to be king. It can be both at once.
Yeah, I'm not going to agree with the perspective that Jon had time to ride back to his men after Rickon was struck. To my eyes, he recognized it was too late to go back at that point. But hey, mileage varies.