A significant issue probably affecting your Yubikey is that there are very few male USB-C plugs available. The ones that do exist are mostly designed for USB-C cables, which uses a tiny 0.8mm thick "paddle card" fully embedded in the plastic cover.
Alternatively, there are "brick nogging" connectors which stand up straight from the PCB and are therefore unusable for a Yubikey, or "sink board" ones which require a custom jig for assembly.
Meanwhile with USB-A you don't even need a male connector at all - just have a bit of your PCB stuck out in a vaguely USB-A-looking shape and it'll work just fine.
Alternatively, there are "brick nogging" connectors which stand up straight from the PCB and are therefore unusable for a Yubikey, or "sink board" ones which require a custom jig for assembly.
Meanwhile with USB-A you don't even need a male connector at all - just have a bit of your PCB stuck out in a vaguely USB-A-looking shape and it'll work just fine.