Troubleshooting Original System 1 hardware is very tough without access to a 100% functional game. If you don't own a second, or don't have access to a friends game, I highly recommend going the 4 in 1 Pascal route.
You didn't say if it responds to any switches (adds some score) when you put the ball in play. A system 1 as a reflex after starting a new game, will energize the outhole coil regardless of if it sees a ball sitting there. So, if that's not happening, you certainly have either a solenoid power issue, or driver board and/or interconnect issue. IF the fuse that is blowing is for the solenoid power, than that would explain no reaction on the playfield (pop bumpers and Sling Shot kickers as well as flippers are self controlled. They only need the flipper enable relay to pass them power to work. They don't require interaction with the CPU other than to register score.)
Based on your description, it appears the start button is at least working. This uses a different connector than the switch matrix to the playfield. But both are prone to Battery leakage damage.