Is this power button compatible with 13th gen?

Ahh, I see.

Unfortunately no, the fingerprint reader does not connect to the mainboard! It actually connects to the input cover, which then connects to the mainboard via a single large ribbon cable (connector). You can attach a power button directly to the mainboard using this Input Connector Shim, which has a schematic on github.

The motherboard does have a small power button directly on the PCB. You can solder wires to that power button, or simply create a mechanism that pushes against this button (this is what the official 3d printed case does, although it’s not the angle you’re going for)!