Looks like the idler gear bore is either too large or the pinion is too small. That pinion is also probably dry, so that would make the movement a little more apparent. I would reccomend you replace both, and sub a bronze idler gear and an offset pinion bolt to close up the backlash. the beauty of a bronze gear is that it is a little more sacrificial and if the alignment is close, it will wear to fit without doing as much damage to the cam or crank gear. If it has a problem and fails, it is also less likley to do damage to the other gears and is far easier to replace than either of the other gears...
You are going to disassemble the engine and look for sand aren't you? Here are a few quick checks for you to do. Reach your hand down into that access hole you showed in your video and into the oil sump compartment section that protrudes from the case(your oil drain plug might be located there?). Run your hand along the roof of that tip-out section and see what you feel or what can come loose. That is one of the more difficult places to get to to clean properly and to paint properly. That is where I found most of my sand inside the case. I also found NO paint there when I tore mine down.
Another easy check is to remove the rod bearing cap and inspect the bearing. If it has scratches, those were most likley caused by sand raining down out of the piston and flowing with the splashed oil into the rod bearing during the test run at the factory. You can clean it all out now, or repair/replace the parts that it wipes out later...
The difference in flywheel ring you noted in your video could indicate a casting flaw, or at least a difference in the castings. It could also indicate a difference in the way the keys are seated attaching the flywheel to the shaft. I wouldn't expect them to be exactly the same, but the significant thing about the ring test is that the tone persists. In a quiet room, the ring tone should go on for 10-15 seconds before you can't hear it anymore, and it should be real difficult to determine when exactly it quits. When I first rang my flywheels, they too were different tones, but one was significantly shorter than the other. Upon further examination, I found a casting flaw/void inside that hub where the shaft goes. The hole I found was about the size of a dime, but the void when probed with a wire appeared to be about the size of a quarter sized sphere(about the size of my thumb). The dealer replaced the wheel. The new wheel is still a different tone, but the tone is pure and the duration is within a second of the other wheel as gauged by my ageing noise damaged ears:) The important thing is also to do this frequently enough to establish a baseline and hopefully be able to detect any significant changes in the future. To this end it might be a good idea to describe the ring tone and duration in your engine log.
Good luck      Â
