there is a great range of ability in UA gate agents. Some are able to crank it out in minutes, others is like pulling teeth.
I would have suggested asking for a supervisor to help in such a situation. Often the GA will want to dump you onto the service desk, but as you saw, you lose all priority and invested time there. I flagged down a passing SD once to get me on my way. You can tell them by the radio and clipboard...
The problem in my case was that at least a dozen flights ex-IAD had been cancelled/delayed that night and the guy who appeared to be the SD was literally running from podium to podium putting out fires. Our situation did not merit his attention.
