[franken]

The starter should essentially be at the temperature of the block, since that's what it's bolted to. Did the block burn your fingers too?
The solenoid is not heat soaked if jumping it (same as adding another solenoid) makes it work. The difference is the amount of current the S is allowed to draw can be more due to the length of cable, ignition switch, connectors, and age of cables. All those need to work right and allow the needed current to pass.
The big ground should be from the battery to the block.

But people don't want the troubleshoot--they want the internet to do it for them.