Yank the Relay Control Module out and have someone go over the solder connections with a magnifying glass. The connections to the "Engine Control Relay", if they are cracked, will cause just what you have described.

But have the person go ahead and reheat and reflow EVERY connection on the board (there are only about two dozen in all, it's a simple board).

If the engine control relay output is not there, then there is no enable sent to the pulse module - and the starter wont' engage.