Personally, I would eliminate the mid-span fulcrum, having the pivot all the way at the far end. The pivot end could then be secured quite simply between two timbers or a notch in a wall timber or so.

You could use a screw mechanism like shown in the video, or more simply just use a block and tackle. Although with a screw on the end you can gain more force.

As for dating press types, there are examples of the remains of screw-type presses, the type with a screw that pushes directly down on the fruit, to be found in the walls of Roman ruins.

Both screw and beam presses are very ancient, with the screw principle known well among ancient Greeks, and the beam principle very simple and straightforward.

I learned a simple trick from Swiss cheese makers and their presses -they call them Schwarpressen. If your beam by itself is not heavy enough, set a big boulder out toward the end of it.