I can't speak directly to the Edge in regards to the battery, but I know my 13 F150 FX4 has given me issues with the BMS (battery management system), in the summer when the AC goes to full blast and the voltage isn't within the threshold to handle the fan motor and the fan shuts off to save power. These more modern cars are so finicky when with these things I can believe a battery caused all of this. Great write-up and troubleshooting steps though! I also replaced our battery in our Edge with an AGM battery, which seems to be a good choice.
I'm curious, what steps did it take to open the gate manually when the actuator wasn't working?