If you need 12V/1A, starting up and showing an error message at 12V/0.2A sounds quite feasible. Of course it depends on what's using up all your power. But at least microprocessors can usually be started at lower power levels (lower frequency) with a switch to high frequency once you've confirmed you have the power available. Display backlight can be dim until you have the power, and peripherals can be powered through a transistor so you can start delivering power after initial system checks.

But it's a bit more involved than just replacing a barrel jack with a USB-C port, and would require some design considerations early on