I think the reason why so many people agree with the "I cannot save everyone fourth oath theory", and why I subscribe to this theory, is because Kaladin swearing that oath forces Kaladin to confront one of his biggest remaining problems. That problem is that he desperately thinks he can save everyone, even though that is completely impossible in warfare. For gods sakes, in the Way of Kings, he actually proclaimed that if he was made bridge crew leader, than noone would die. That conclusion alone suggests that he is completely in denial, and that he puts far to much pressure on himself. He needs to accept that this isn't his fault. I also know that in the Urithiru gem archive, a novice Windrunner says that:
"My spren claims that recording this will be good for me, so here I go. Everyone says I will swear the Fourth Ideal soon, and in so doing, earn my armor. I simply don't think that I can. Am I not supposed to want to help people?"
This seems to indicate that the Fourth Ideal ostensibly involves "not helping people" which seems to fit the "I cannot save everyone fourth oath theory". You said you think evidence points against that theory. I am curious what you mean by that.