USB-C continues the "2 steps forward, 3 steps back" of USB history.
A USB-C supply (a box with a wall plug on one end and a USB-C socket on the other) defaults to providing no power. This is because all USB-C to USB-C cables are supposed to be male at both ends, and you can imagine the hilarity that would ensue when two 60W+ supplies both tried to power the same cable.
In order for a USB-C supply to provide power, it either needs to see "programming resistors" on its CC1 and CC2 pins that tell it what sort of thing it is connected to*, or an intelligent IC that can negotiate for higher power, current, or both.
* There have been various unintentional and intentional foulups along the way. The first production rev of the Raspberry Pi 4 said "I'm a pair of passive headphones, don't worry about giving me any power". Certain brands of smartphones/tablets ship with chargers that default to wildly inappropriate voltage and/or current output (one such supply provides over 20V). In theory, to comply with the USB Implementers Forum requirements, devices that specify their desire via programming resistors are limited to the "classic" 500mA at 5V. Raspberry Pi (or other large resellers like Canakit) provide supplies that can provide far more than 500mA (which won't get you far on a modern Pi), but which apparently do not utilize the CC pins. All of these Pi-specific chargers that I've seen have had USB-C cables permanently attached to the wall plug part.
A USB-A charger with a USB-A to USB-C cable doesn't know anything about any of this, and just supplies 5V, presumably at the 500mA standard.
In summary: A USB-C charger and a USB-C to USB-C cable won't work with the snap-in 2-wire socket option on the back of the case. A Raspberry Pi (or other large reseller) charger will probably power it, but the leads I've seen on those snap-in sockets don't inspire confidence that they are sufficient for the current. A USB-A to USB-C cable into that socket may provide enough current to start the PI, but probably not.
Users are much better off using a Raspberry PI (or other major brand) Pi supply plugged directly into the USB C port on the back of the PI. After all, it is sticking out of the back of the case on the PiDP-10 (unlike on the -11).