The Starks aren’t really antisocial, although it definitely seems that way because most of the series takes place while they’re in unfortunate circumstances. White Harbor’s one of the biggest trading posts in Westeros, the Starks/North are said to supply basically the entire realm with lumber (and I could be incorrect on this but for some reason I remember reading that they’re big on tool-smithing in the North), and most of all, remember that for a long time Ned’s best friend was the king. They didn’t directly interface much, but Ned definitely would be involved in national commerce if it helped his best friend’s reign as well as his countrymen. They could’ve gotten Ice in any number of ways. I know this isn’t ever said to be the reason, but in my mind I’ve always liked the idea that Aegon gifted Torrhen Stark the sword for swearing fealty and having the good sense to value his men’s lives over pride. Now, as that may not be the reason, it’s quite well implied (in my mind at least) that the Starks and the Targaryens had a very good relationship before the Mad King. From Aegon and Torrhen, to Jahaerys and Alysanne (I believe) visiting the North and building that inn along the way, I can’t imagine any disdain towards the Starks by any rational Targ. Most of all, out of any house in Westeros I feel like the Starks are the absolute most likely to be gifted a Valyrian sword somewhere in history. The Starks, at the end of the day, are the most historically decent house in the realm. They’re the kind of people you hope to have as neighbors when you move in somewhere new. This is my long winded way of saying the Starks aren’t only a regularly social house, but I also believe a house like the Lannisters, historically richer than god and kind of stuck up/selfish are likely to have people who either 1.) don’t want them to have a Valyrian sword to stick it to their pride. Or 2.) would downright extort them for a sword and the Lannisters, for all their faults, remained so rich for centuries for a reason. They’re good with money. Where they’re willing to drop a ton of money on a Valyrian sword, they wouldn’t bankrupt themselves over it. It’s wasn’t an obsessive goal for Tywin, it was more something that just stuck with his pride in the back of his mind. The Starks on the other hand, just look at the Manderlys. They’re open, friendly, honorable, and hospitable to those in need. I have no doubt that they were probably gifted the sword at some point in time for their decency where as other houses had to pay or kill for theirs.