Chemically, it takes conditions of pH over 6 and heat over 170?F (or there abouts) to solubilize tannins. Pressure could result in lowering the heat requirement, but I don't know (nor have I looked) at any research on that topic.
When draining the wort from the grains, the first thing you do is start the drain slowly. This allows the husks and larger kernel particles to form a filter bed through which the finer particles will get trapped. The presence of hulls is the real key, as they act as channels for the wort to flow down through the every increasing depth of spent grains. Shred the hulls too much, such as in a congress mash, and there is no chance of making a filter bed to drain the wort efficiently.
When you drain the wort too fast at the start, it compresses the filter bed quickly and removes the void areas which allow the wort to flow.
Now, go back to the french press idea. The filter is on the top, where most of the finer particles will settle. Press a filter or fine mesh against it and apply pressure and you will quickly compress the finer particles against the filter, thus cutting down on the void areas and ceasing the flow of wort.
A cider press works on a different principal. There are gaps between the boards which allows the juices to flow out through the sides as well as the bottom. The grain of the wood also helps in channeling the juices away from the pulp.
As someone who does mostly BIAB, most people allow some or most of the wort to drain first. When you twist of squeeze the bag, there is areas along the whole surface of the bag for the wort to drain, so it takes the least resistant path out of the grains and bag. Oh, and yes, the bag can be blinded over and stop wort flow out. I had one early recipe of 50% rye malt (hull less) and flaked rye which I started squeezing much too early and ended up with a gummy mess on the outside of the bag and missing my kettle volume by a fair amount due to the entrained water left in the grains.
I can think of a number of reasons why it might not work, but the proof is in the results. By all means try it. As someone who works in product development, you never really know until you try if you missed something or not -- it just boils down to probability of success versus effort input.