I get the impression that they do it a bit differently in the north.

Actually the way things work in the north would have worked perfectly fine in the south, just Ned uncharacteristically did not act at all how we know he ruled the north.

Look over all the info we get from Jon, Robb, Catelyn, Arya, Bran and Theon. Jon's been to Deepwood Motte and the Dreadfort. Robb's been to the Dreadfort, and says that Ned taught him to get to know his men so that they didn't have to die for a stranger. Catelyn says that she met Jorah and his wife dozens of times. Arya says she's been to White Harbor 3 times. Bran says he and his brothers and the Cerwyns were always close, that every harvest the Starks throw a feast for all their bannermen, and that every night Ned picks out a random person and tells them to sup at the head table so that he can get to know them and their work better. Theon's been to Torrhen's Square. Etc. Ned clearly visited/hosted his bannermen, and made an effort to know them ENORMOUSLY. He was constantly going around the north, checking in on his bannermen, and keeping up with them. He then gets to King's Landing and... nothing. He doesn't check in and find out who's friends with who, what power Renly holds, what Littlefinger does, who among the Kingsguard are trustworthy, etc. He just accepts everything at face value basically (beyond noting how powerful the Lannisters are), when we know that he didn't do this at all back home. It's out of character.

Secondly, we also know from Robb that Ned taught him that the northern lords only respect strength, that you needed to submit them to your will or they'd walk all over you. Ned clearly submitted them to his will back in the day as they were all loyal, and still are, to him, and from the above we know he kept up that if he was always checking on them and basically reminding them who's their lord, but again, he just abandons this when he gets to King's Landing. He's the Hand of the King, and brought more personal guards than anybody else had in the city, yet he never throws his power around the way you'd expect from a man who rules his own lands by an iron fist. No visits to shakedown Littlefinger when he's being a dick, no slapping Renly around to be serious, no nothing. He's the second most powerful man in the kingdom, yet he let everybody walk all over him, when from what we know he should've immediately tried to seize power when that's how he knows to rule is through power.

I mean look at what Cregan Stark did back in the Dance of the Dragon. He just walked into King's Landing with an army and declared himself Hand of the King, for a man he hadn't even agreed to fight for, ruled for seven days, sentenced a shit ton of people to die, and then just left. Nobody questioned his authority, and he acted exactly as we're told the Lord of the North acts.

It's Ned's uncharacteristic actions that get him outplayed. If he'd acted as we know he should have based on how the north is ruled, and ruled by him no less, he could've been Hand just fine. Instead everybody saw weakness in how he was acting, or more importantly not acting.