Everything, which has worked sometimes, can work again. Software calibration is necessary only if the gimbal controller is brand new and has never been used in the working camera. Often repaired cameras didn't pass camera calibration on the drone but they're fully working in all other aspects.
The magnet is completely missing in those two pictures. I assume you had a ball of fuzz and wiped it away? You will need to find a magnet somewhere. If you have a crashed camera around it would be great. Otherwise you might want to contact John Mitchell at https://yuneecskins.com/contact-us. John may ask you send the camera in for repair.
The "long,long.short" beep is just the encoder error code. That part is normal when the magnet is missing. But it would normally continue without more than the short pause between cycles. Yours seems to have a significant pause in the video, suggesting something else may be going on.