If the projector is mounted overhead then you are left with a bundle of plugs, likely none of which USB-C.

The situation with display connectors is particularly sad even without USB-C, as we went quickly from still largely VGA and maybe DVI a few years ago to HDMI/DP/mini-DP co-existing, and now USB-C in the mix. I just recently had to buy a USB-C to DP cable even though my work laptop has HDMI and I've been using it fine, becaues this particular monitor does not support anything bigger than 1920x1080@60 over HDMI, and DP is only offered via USB-C on the laptop side. How often do you see this particular combination being required a year ago? But here we are, dealing with a quasi-N^2 problem.