Not Getting 12 Volts at Coil

The remote switch you have is simply a mechaics helper that gets tagged onto a starter relay or sylinod switch to engage the starter and rotate the engine. For the engine to start and run the ignition switch must be in the run position.
You could purchase a new ignition switch and connect it at the factory harness connecter instead of fully installing it. Turn it by hand. The switch would be grounded when installed but isn't depending on a ground to make circuits in its various positions.
Another simple test would be hook up a good battery charger and set it on 50 amp boost. That would make the entire system hotter.