There is a potential weak spot in the original battery cable routing of a '57 that may be
part of the problem.
There are 2 large cables that provide (+) power to the soleniod & starter, and two more large diameter cables that complete the Ground half of the circuit back to the battery from the starter.
All eight ends of these four cables must have clean and tight connections for the system to work well.
The often unnoticed 'weak link' in these cables is poor connections in the Ground cables for the body, engine & starter. The Negative battery cable fastens to the firewall just behind the battery, and 4 inches to the right of the
master cylinder another negative/ground cable connects the firewall to the engine.
This detour of the negative battery cable to the firewall before it connects to the engine & starter adds extra spots where corrosion can develop relatively unnoticed.
Due to the high current needed to turn the starter and engine all the connections must be clean (metal to metal) and tight to work their best.
I'm assuming the engine compartment was repainted some time in the past as the black color doesn't match the exterior of the car. The extra paint may interfere with the starter solenoid being electrically grounded to the firewall. This can prevent it from working as it should.
In your 5th photo I'm seeing a clipped off red wire that could have connected to the solenoid and a White wire that would have originally been Red with a blue stripe, fastened to the S terminal.
12v power to the S terminal is what triggers the starter solenoid to supply power the starter. Diagram attached below...