1971 Dodge Dart Swinger Turn Signals and Reverse Lamps

power to the flasher as soon as the key is turned on.
Yes. To the flasher unit. It's a long route. On the factory diagram follow wire X1 to B1 to D1. (B1 is power to the back up light switch.)
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If the connector or harnesses to the reverse light switch has issues (like grounding to the case) then the fuse will blow, or at minimum there will be no power to the turn signal flasher because it found a shorter route.