I'm not personally upset about it. I just think it doesn't make sense in the narrative of the show. I strongly disagree that her way was "the least unreasonable." She has the Stark name, and that's the root of all her power. Littlefinger knows that, the Boltons especially know that, she knows it. I'm not saying she raises an army and leads them in battle. But she could go to White Harbor, tell the Manderlys what's up, and the Manderlys would fight the Boltons to the death. And that's just one example of the Northern houses.
I think you're a bit too dismissive of how much power Sansa has as the "last Stark" (as far as most people know), and the visceral dislike of the Boltons, even before there was ever a Red Wedding. And Littlefinger would have known Stannis would rally around her, he just wanted her in Winterfell for his own ends (the show Littlefinger seems a lot more self-interested and less obsessed with Sansa). If she could get to Winterfell from the Vale, she could get to the Wall from the Vale.