Do controllers draw more power than keyboards? I wonder if, in spite of upgrading the power, there is actually a power issue. I'm not sure why the power symbol isn't showing if that is the case though.
Could be, usually a keyboard/mouse need about 100 mA, while a gamepad may require more.
There was a topic a while ago (see here) that surveyed the power usage for various controllers. It has the command to query the power the controller reported as needed, you can run the command and see what's reported by the Xbox controller.
At one time, there was a max_usb_current option for the PI (in config.txt) which increased the power offered by the USB port (600 -> 1200 mA), but now it's on by default and so it's not needed anymore.
Is it possible something I copied over from the previous Pi isn't playing nice? I thought I only copied over saves/emulator specific settings, but in the absence of anything else I wonder if something is getting in the way.
Highly unlikely - flashing a new image showed it's not that.