Another timely thread for me as I have a similar issue.
The lead acid battery of 1980 with its 20 amp capacity was only just good enough when the bike was new and everything working properly - give them a few years -carb problems start to show up and is it any wonder so many of these bikes were pushed into a corner and forgotten.
Many owners , like myself, purchased the Odyssey 680 with it's 170 cca. A huge step forward from the original. We were all quite happy for a while, having overcome one of the bikes weaknesses.
But once I installed the big bore kit ,a somewhat similar problem showed up as mentioned by you other 3 guys. Hot engine, stop to fill up with gasoline , attempted a restart and the engine appeared to be "locked". No clicking of starter solenoid, battery at 13.5 volts, but push the starter button and nothing. I have re-wired the whole bike, installed a larger starter solenoid, swapped out several starter motors but nothing has changed.
Several early model Goldwing owners have told me of similar problems -essentially the motor stopping just in the wrong position when stopping for fuel for instance, And then the electrical system not having enough oomph to turn the engine over. I don't know whether that is correct or not, but in my case, I would pull the clutch in, make sure the gearbox was in neutral, and after several more pushes of the starter button the engine would start - but the peculiar thing with that is , the engine wouldn't even turn half a revolution before it would fire.
Solutions : Most people would be using a modern battery with huge power output over the original. I have a 270 cca lithion -ion battery on my MV Agusta but that didn't help the Kawasaki .
Peculiar to me again, put the gearbox into 2nd or 3rd gear and rock the engine past the "locking point" - haven't had to do this yet.
As Neville mentioned - the starter motor , like the oem batteries is not really upto the job.
Several years ago , new starter motors became available - they were around $100. Then nearly 300 and now...……
canadianautopart.com/new-mitsuba-starter...yvgGfEAQYASABEgLLl_D
This is a vey good price. Amazon has them for nearly $300
Replace the battery ground wire. Only old piece of wire on the entire bike.