There's only so much you can do to diagnose these kinds of faults without proper test equipment. I know you've tested for continuity, but that really doesn't tell you much here.
To do this safely, at minimum, you'll need a logic pulser and a logic probe - You'll want to use the pulser to 'inject' a pulse into Pin 40 of Paula, and use the probe to look at the TXD pin of your serial port.
If you see a pulse at the TXD pin (it'll be the opposite polarity of the pulse you injected, as it'll pass through U304 which is being used as an inverter), then you know the entire path between Paula, through U304, through EMI327 and through the serial port header is fine, and it's likely a software fault.
If you don't see the pulse, you can backtrack to the input side of EMI327 and try again. If you still don't see the pulse, look at Pin 8 (output) of U304. If you *still* don't see the pulse, look at the input of U304 (pins 9 & 10, which should be shorted together)
There are ways around the use of the logic injector/probe, but if you're inexperienced with these kinds of things, you might not want to risk damaging your Amiga.
