Honestly sounds like a loose connection somewhere and not a broken camera.
The easiest thing you can do is pop out the plastic trim piece on the inside of the tailgate and double check the wiring & connections on the wiring harness inside the tailgate. The camera is powered whenever the vehicle is on, so turn on the Bronco and check to make sure the camera is begin fed voltage from wiring harness.
Hopefully that's where the issue is. If it isn't there... the next thing I would is hook up the vehicle to a laptop and see if the APIM or IPMB is throwing any errors. If it isn't... it's going to be painful... you're going to have to trace the wiring from the camera the way back to the IPMB (Image Processing Module) and/or APIM (Accessory Protocol Interface Module) to see where the issue is. Which means you'll be pulling trim pieces and ripping out a lot of your interior. (This is likely why your dealer is asking to hold your vehicle for approx 2 weeks.)
2022 4dr Badlands, 2.3L Manual, Mid pkg
YouTube: youtube.com/@ragnarkon
Instagram: @ragnar.kon