unless you need the boat right away or do not have the time to do it, it looks like your case is doable, be patient and do not guess...been there myself and wasted a lot of money...I am not even close to be a newbie in mechanics, I just happen to buy a boat with a Johnson that had more problems than a book of physics....decided to give it a shot and started reading..... these guys in here are the real thing when it comes to knowledge and kindness not like other sites....i..and by asking for help here and reading I was able to bring it back to life...although it looks like it still needs some adjusting but it works now....my point is, do not get frustrated, follow every question Fred, Jerry and other ask and try to respond in line, you will see....
as I said not a mech here but have a lot of electrical/electronics background...your issue looks very similar to what my cousin had with his engine and I was able to help at least finding out that after the engine would cutoff it was not pushing juice through that "module" (powerpack), the engine needed to be at certain temp below that cutoff limit for it to start again...powerpack it was....it was a pain to troubleshoot it on the water but we identified that no pulses of voltage were coming out the module when engine got hot even when the stator was feeding properly.....if the shop do not fix the problem these guys in here will....just be patient .....
by the way I ended melting down the epoxy out of the powerpack just for the fun of it and see what was inside....two rectifier diodes were having leaks which increased with temp and caused shorts....