What could be the problem here?
The last time you successfully used the flash may have depleted your batteries. Flash uses a lot of power compared to the rest of the camera. Either install fresh batteries or check the voltage of the old ones.
After that the LCD will work again but shows an empty battery indicator, although I'm quite sure the batteries weren't empty.
This is normally an indication that your batteries are low. Either install fresh batteries or check the voltage of the old ones.
On page 53 of the EOS 500/500 QD Instructions the first item listed in the Troubleshooting Tips is:
No display on panel
The two suggested causes are:
Are the batteries exhausted?
Are the batteries the wrong way round?
The solutions listed are:
Replace the batteries
Reload the batteries correctly
Before attempting any disassembly of the camera, I'd get two fresh CR123A batteries and test the camera with those.
If that doesn't do the trick.
- Try prying the built-in flash open while the whining sound is heard. If successful, check to see if anything could be preventing the flash from popping up on its own. The hook that holds the flash closed (against pressure from a spring that opens it when the hook is released) may not be moving properly.
- Place an accessory shoe cover on the camera's hot shoe. This will disable the built-in flash. There's a small switch located under one of the friction clips of the ground rails on either side of the hot shoe that the camera uses to detect when any accessory is inserted into the hot shoe (it's on the left side for every Canon camera that I've needed to clean/check the switch, but I suppose it could be on the right side for some cameras).
- Place an external flash in the camera's hot shoe. Any manual flash should disable the built-in popup flash, but if the hot shoe detector switch isn't working, a Canon "dedicated" (what we now call 'TTL') flash that is powered on should also disable the popup flash.
Update from the OP
Looks like I was a bit quick with the question. A new set of batteries did resolve the problem! I'm surprised that they depleted so quickly given that not many rolls had been shot with them and I only used the flash to test it and maybe on two photos or so. I'm gonna voltage-check the old ones. Nice tip on the accessory shoe cover, that could get me out of a flash malfunction in the future.