The question is why you had wasted precious time with the CS desk and the unit manager in the first place. People at the airport are busy with making sure the flights depart more or less timely, which is, quite honestly, more important than just keeping some EXP happy.
You should have called the EXP desk, where the supervisors can make things happen a lot easier, and should have asked for guaranteed availability in the main cabin with a waiver of the 24 hr advance res requirement. Then they would likely have IDB'd someone on the next flight and would have given you a confirmed seat. Or they could have simply accommodated you OAL.
I've had this done on US. If you're top tier, you generally get a lot better results via the elite line than you possibly can with the airport staff - because the elite line super can call GA and direct her to do specific things in real time, which you will never get out of GA by yourself.
It doesn't always work that way. Sometimes it's better to work with someone who actually has authority and is right there to get things done - especially if you already have their ear. And there have been times where I've called in and they've told me to deal with the airport as the flights are under their control. Lazy rep, who knows? But it happens.
What you said is true generally, especially when it comes to standing in line at CS vs getting on the phone. That said, it isn't always the silver bullet you're making it out to be.