Ya know it might be a simple thing. The fact that it runs only with your foot on the throttle suggests that when it comes off the fast idle motor, the throttle opening is insufficient to feed idle air. Given that there is no fuel connection to the throttle it certainly isn't a carburetor's choke and accelerator pump that keeps it going.
This is not to say that there isn't a problem with the temp sensor or the throttle position sensor. But I'd get the manual and make sure the throttle adjustments and the fast idle motor are working and adjusted properly.