The next time the car won't engage the starter, just let the key relax back to the "on" position, wait 30 seconds.
Does the engine fan warp up to high speed (you WILL hear it)?

If so, it is 99.9% likely it's the RCM. In that case, yank the RCM (no tools needed, it unplugs) and send it to D.J.; he posted above and is still active on the forum.

If the fan does not spin up, well, we have more homework to do.