I believe the self timer will do that if the shutter is not cocked (vid 2). In terms of vid 1, it looks like the shutter won't latch in the 'cocked' mode.
Important info would be if the camera has ever been taken apart. If it has been taken apart prior, then it really could be anything and may be totally ruined. If it had never been apart before, look for a small lever that is frozen or gummed up on its pivot. Also, a pin may have broken off.
If it has never been apart before, there is a high chance that a flush with appropriate solvent will get it working again.