Yes it's possible. I would argue it's not a good idea though, the existing transmitters for flying things generally don't have the left stick/gimbal automatically re-center with a spring load like the xbox joysticks do. With quads 0 throttle is stick all the way down unless you are flying 3d mode but that is atypical, with 3d mode the center stick is 0 throttle and down is more negative throttle up is more positive throttle.
In general what you need to have a control link is a transmitter and a receiver. The transmitter/receiver pair establish a link over the air with radio waves, you could use something like an rf-nano or arduino-nano+nordic rf power amplified transmitter on the ground and an rf-nano as the receiver on the quad, then on the receiver side you take whatever control signals are coming in over the air (from your transmitter) and you parse out the data and send the appropriate control signals to the flight controller. That last step of translate/send the signals to the flight controller would probably be the most challenging but if can find some libraries that will output sbus or CSRF or PWM signals that betaflight can "understand"/interpret then you can make your own control link or otherwise send in control input from some outside source and let betaflight handle the actual flying (PID loop etc.)
That said the flight controller is itself a little computer just like the raspberry pi, so if it has enough UARTs available you could just about as easily hack the betaflight source code to patch it with whatever additions you would want (assuming not pushing the STM32F7 or whatever is on your FC above it's limits with regard to compute/memory).
