First, the northern lords don't know that; it happened in king's landing with all the other northerners killed.
Second, the war was started by Catelyn taking Tyrion prisoner. It would have been impossible for Lannisters not to retaliate. Eddard says it was at his command to protect his wife.
Third, why the fuck was she called "the smartest person I know" or "smartest girl I know" a couple of episodes back? She has been an air-headed bimbo for most of the show ("I don't want to marry someone smart and brave, I want Joffrey!") and even after her supposed transformation the intelligence of her moves have been a liability (didn't tell jon about the Vale support, only outsmarted Littlefinger because he suddenly had a plotbotomy, supposedly could see through Dany, but then her actions, just telling Tyrion, mostly depended on others doing the thinking for her).
Considering there are 1 or 2 stark male claimants, it seems far more likely for either of them being put on the throne. I can only conclude that she's on the throne because of muh feminism in both writers, and GRRM. Yes, I think he would do something like this too. But perhaps we'll actually experience her elusive intelligence in the books he's never going to finish.