
IT'S WORKING!
Backed up player folder. Uninstalled Enigma. Manually removed any references to "enigma", "tesseraction" and "dreamcatcher" from the Windows registry.
Rebooted. Reinstalled. Rebooted again. Entered CD key. Went into preferences and ran the mike test. I had to lean close to my built in mike but it worked!
Installed the US Gold upgrade patch. Rebooted. Restored my player directory. Started ERT. Agreed to EULA.

Went into preferences and tested mike again. Works!
Started a patrol mission and barked some orders. Worked!
Played another 2 minutes and crashed to desktop.
I'm sure that was a fluke. At least I hope it was.
UPDATE: The crashes continued until I lowered the audio card's hardware acceleration setting from maximum down one notch. I had upped it yesterday during my testing of the SAPI problem and forgot to lower it. So, if you're having crashes, lowering audio card hardware acceleration might resolve the problem for you, too.
Now I'm whistling happy like Steamboat Willy!