I'd be looking in scrapyards for two rack and spur pinion sections (conventional spur gear type) from the carriage drive on a long bed lathe or similar, if not long enough, use two racks back to back.
Really crude but would probably work OK for a trailer push out as in the video is a length of roller chain welded to the bars at intervals, synchronised by two chain sprockets on a common shaft. Again these things can be found at scrapyards, but old chain and sprockets are often thrown away at motorcycle shops. If you went to a local motorcycle shop they may have some or will put some aside for you, maybe free, likely very low cost if any.
The ones in the video look to be homemade from a steel drum and some cone head rivets.
Good luck JD