From your pics it looks like from the 12v switch you are stepping down in 'wire' size to the push button switch. You go from yellow crimp on connectors to blue. A yellow gauge wire shouldn't fit in a blue connecter, looks like you have stands not crimped in the blue ones. What happens if you manually connect the 12v switch to the starter wire, basically bypass the push button? Look at your voltage drop. If good I suspect the push button or wiring to it. If there is a voltage drop it's down the line, started, battery etc. eliminate/isolate is the easiest way to minimize things.
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