It should be a double lever, double set type which allows it to fire either way - but seeing the downward step in the front trigger bar it may not be. Without having it in my hand and placing it back into the stock to see it's relationship with the sear bar - I can't tell. Either way, there has been some good points given. The set screw on the bottom between the two triggers needs to be backed out. The front of the rear trigger where it engages the catch on the front trigger does seem to be rounded also and that could be the reason it is not holding. W/O the spring in there to apply rearward pressure on the front trigger - the video you show would not work properly anyway.
An experienced builder could diagnose the problem very quickly if they had it in their hands. It would be very easy to put a new trigger in the rifle - either a double set type or a single one.
