Sounds like they have added a solenoid and a start button/switch and kept the original lever as a neutral safety feature.
Keeping the original lever is a good thing, but why have both? The original contacts are capable of carrying full starter current, but were troublesome. If you want to keep it original, the solenoid could be simply eliminated. Or keep the solenoid, remove the start button, and use the original lever to actuate the solenoid.
Most likely the contacts on the start lever are dirty or out of adjustment. A simple cleaning and adjusting would be a good place to start.
Also check the battery cables for clean and tight, check the cable connections, check the ground cable. On a fresh restore, sometimes the paint works as an insulator under the ground cable or between the starter flange and the engine block.