I saw a Twitter thread where they explain another reason why you should never connect the device to a PC:
As long as I know Jade is the only one able to do this.
The Blockstream Jade requires some type of connection to a PC when you first set it up to install its firmware. You can install the firmware either via the Blockstream Green software wallet or from Blockstream's download portal. There is a third (advanced) option that I have forgotten about. You can connect the device either via USB cable or Bluetooth to install the firmware. The same is true for firmware updates, with one difference being that you can install a firmware with disabled Bluetooth. If you do that, then you can only install new firmware updates by connecting the device via a cable.
You are right on one point here: having a signing device (a hardware wallet) and using it via Bluetooth is nonsense, given the amount of bad code an attacker can infect the Bluetooth stack with.
Also the attacker.