How about some pictures of the gear selector and associated parts? When I first read your thread, my guess would have been to start there. If that key broke and has been rattling around in the crankcase, it would certainly cause issues, but I'm not sure why the issue would always be first gear. Was the key partially in the crank shaft still?
From what I can see in your videos, the transmission gear teeth look to be in good shape. How about the shift drum that lifts and drops the shift forks? This is the cylinder in the middle with all the shift patterns that engage the shift teeth.
Check out gears 10 and 20. These are the two that are meshed and under power when in first gear. Also inspect the pockets and inner engagement teeth that fit inside the gears when the shift forks slide them up and down and engage first gear.