The Quadstick is always powered via the USB-B port regardless of which port is used for the host data connection. The USB-A port supplies power to the device that is plugged into it, but it is not intended to power something like tablet. It was meant for other low power basic devices like joysticks or controllers. Most tablets or phones have OTG USB hub accessories that can power the tablet/phone and the things plugged into the USB ports.
A quadstick by itself will not work on a phone or tablet for remote play. You did not mention the target console, but would probably need a CronusMax Zen or Titan Two in any case. See their websites for the details.
There is a new device called the
Beloader that uses remote play to get around the security of the PS5, but you would need to directly display the console on a monitor. You could not simultaneously remote play using the Beload and a remote play app on another host for the display at the same time.