EDIT2: The drivers sure were up-to-date, but apparently the BIOS wasn't. A BIOS flash to the latest version and the connection works perfectly. Problem solved, thanks for the help!
EDIT3: Oops, the solution wasn't the BIOS update after all, but instead the fact that I plugged the laptop into the AC adapter for the update. Apparently the USB-C port only functions when the machine is being charged for some weird reason...
I have the latest drivers on the laptop. I did try manually selecting the USB-C as the input, but there was no signal detected. The cable is some generic one, so I think it's impossible to tell whether it supports video or not. It's probably this one, although mine's shorter: https://www.ebay.com/itm/Super-High-speed-USB-3-1-Cable-E118077-7-AWM-21572-80-C-30V-VW-1-Ji-Haw-QTY-10-/163610532488
I tried googling for a video capable usb-c cable though and the only ones I could come up with were some that have long since gone out of stock. Is video over a type c to type c cable not a thing at all? Should I instead spring for a type c to HDMI 2.0 adapter? The monitor only has one DP connection, and I'd rather keep that for my desktop PC.
EDIT: To clarify, the laptop uses the integrated Intel HD 620 for external monitor connections, and according to this page it supports HDMI 1.4 (which presumably is the HDMI connection) and DisplayPort 1.2, which probably would be via that USB-C port. And as DP 1.2 can support up to 4k 60Hz I'm hopeful I could get the monitor running with 3440x1440 60Hz if I was able to procure a signal from that.