Try saying that 5 times quickly...

I'll always sorta knew it, and I've often used that technique to resolve bookings, but mine have always been changable tickets. That is, if the web site doesn't not display the flights I want, I book something else then call back and change to the flights I really want. Did just that, just now.

Only this time the agent took the time to explain to me that now, even on non-refundable, non-changable tickets, if you call back within 24 hours of making the booking they will happily change the itinerary to the flights you want (presuming there is space available in your ticketed class.)

That's kind of a double whammy bonus, as you can make the booking, but cancel it if your other arrangements don't work out or something changes in the next 24 hours. But it also means that the fact that the web site often shows silly routings overlooking better ones is now easily overcome. It's an extra step, but it's free and adds a lot of flexibility.