How do I wire up an aftermarket USB Type-C connector so that it works?
The White and Blue are not D+/D- wires. They are likely CC1 and CC2 wires. Try to solder them as such to your breakout connectors if you want the charger to operate in standard functionality.
If you want the charger to just output the default +5V, try to pull down one (or both) blue/white wires with 5.1k resistors to ground. Type-C ports/chargers enable default power only when they detect a presense of a device, which is 5.1k to ground on CC pins.
CORRECTION: from your picture and looking at the relative wire gauges, it looks like the White wire is VBUS power, and Red and Blue are CC wires. And yes, if the charger is not dead, red or blue wire should have some weak voltage source on them, to detect CC connection.