It depends. If it’s a smaller airport that isn’t served by any of Korean’s partners, then there wouldn’t be any awards possible and they might not bother to add that airport to their tool.

If you’re sure that the airport is served by a partner and that an award should be possible, then you can call Korean to book by phone. You might be able to search for award space in advance before your call using another partner like Delta or Air France.

Sometimes an airline will give more award space to one partner than another. There are unclear reasons for this, ranging from an intentional decision to technical issues. Two examples are how Singapore Airlines limits the award space it provides to United Airlines and how Emirates has intermittent issues providing award space to Alaska Airlines.

However, I’ve never heard of a case where an airline gives less award space to its own customers like you describe, where BA gives more space to American Airlines than to BA Executive Club members. A more likely explanation is a delay in synching availability between both systems, so that the space on AA’s site didn’t really exist, or a technical issue on BA’s site so that not all space was being properly displayed. In either case my recommendation is to call the airline when you can’t book what you think should be available.