If all that mattered was being on the same network, it doesn't matter whether that's WiFi or Ethernet. Frames got switched between those.

WiFi Direct uses, as the same implies, a direct P2P network between the two devices without a third network device in between.
That would indeed not work if the device has no WiFi adapter.
You may perhaps be able to remedy that using a USB WiFi adapter supported by the mainline Linux kernel version running on your device.

Have you tried it though? I'm pretty sure Miracast works over the network too.