Originally Posted by
username
My experience is that airport agents are a lot more flexible than phone [1k] agents (or it could be that I am just a lot more charming in person)

I have always dealt with the United Club agents when I have missed a connecting flight. I have always been polite and friendly, and have suggested preferred alternate flights. I have always gotten the flights that I have wanted, often on different airlines, sometimes with minimum connection time rules waived (when I knew I could make the connection), and nearly always with a smile from the agent.
Hope I haven't jinxed it with this post.