Why is everyone debating some theoretical advanced heartbeat or otherwise people detection tech rather than the absolutely obvious answer - some kind of advanced, specialized transmitter that's designed to be hard to detect and simply transmits the encrypted GPS coordinates of the pilot?

Because the NY Post ran an article that said

"The CIA used a futuristic new tool called “Ghost Murmur” to find and rescue the second American airman who was shot down in southern Iran, The Post has learned. The secret technology uses long-range quantum magnetometry to find the electromagnetic signal of a human heartbeat"

Note, I agree that it was probably some novel beacon technology. Just answering your question about why people are debating whether it was a device that could detect a human heartbeat from long range.

Or just rocking up to the nearest village with a thousand dollars in cash and asking where the pilot is.

I think the fear of being located isn't based on the fact that someone can decrypt an encrypted transmission, its simply because someone can trace that a particular location is transmitting some radio waves.

That's a complicated way of describing Professor Xavier's Cerebro, but that's basically how it works.