I don’t think so? You covered what if he decided to wait even though Sansa was also a captive – given that they were too valuable to kill – and the possibility of him actually taking the Black, including that he might not consider such blatantly Faith-based oaths of silence binding, and would be with Arya besides. But I don’t think you brought up the only Stark captive option. (I may have reread your stuff a lot, your analyses are useful for AU worldbuilding and generally intriguing besides.)

Ah ok. Well, if Stark is the only prisoner, a couple things happen:

1. Ned’s not confessing. He only did that to save his kids. Now, he’s not in good shape atm, so it’s still possible he dies in his cell, but that probably butterflys away his death at Joffrey’s order.

2. There will be a strong push to trade Ned for Jaime, and that push will have support from Robb’s bannermen: trading the Kingslayer for the Lord Paramount of the North is an equal trade at worst, and arguably a trade-up for the North. 

3. It’s highly unlikely Robb becomes King in the North. An alive Eddard is going to reveal his investigation and push for the North and the Riverlands to back Stannis as the rightful King. 

4. This in turn changes Stannis’ strategy in both political and military terms. On a political level, with two kingdoms behind him, Stannis doesn’t seem like a no-hoper, and with the Hand of the King and Protector of the Realm legitimating his open letter, you’re going to get people who were on the fence shifting to Stannis’ side. On a military level, Stannis can take his men up to Maidenpool, unite with the Starks and Tullys, and make a hard push on King’s Landing from the land, negating most of Tyrion’s strategems, before Renly is anywhere near the capitol.