I think the sound in the video is similar to the sound in the audio clip, so I still think it is caused by something being stuck or having too much friction and so causing gears to slip.
Since my last post in this thread I salvaged a Canon MG5350 and
after a little cleaning it was OK. So I searched for a service manual but found none. But the MG5300 series is mechanically similar to the MG5200 series so I think an MG5200 service manual will also be useful. It has got a good disassembly instruction and apart from the top scanner parts the printers might be identical. You find working links to that manual and a collection of photos from dismantling a MG5350
in this thread.
Instead of gears slipping the noise could also be caused by a clutch not properly engaging,
See this and
this post.
Good luck with the troubleshooting. I think the MG5350 is much better than its successor the MG5450S.