To my understanding the influence of all joints to one point should be 100% total. If that is a valid point, then the file you have attached has a problem. The folding map has the need that at one time one point is controlled to 100% by one joint and then later on by another one. This needs to be solved.
I check quickly your weighting and I saw that not all joints are weighted nor that each one is completely weighted. So, there is another problem.
However, having gotten the paper folding question in the old forum from time to time, I know that there are other problems. I personally wouldn’t use Joints for an project like that. I even would consider to split the whole animation in two states, the part where it un-folds in one direction and then a new one to the other direction (following the first seconds of your movie link (Which I had screen captured to see it image by image)
Another way is to use the PoseMorph. I have attached a small little sketch, well knowing that the folds needs more attention, but I think it transports the idea. Use perhaps two fold (point/edges) to get the thickness of the paper established.
Other examples can be done with the Cluster tool, Andy did a folding Menu card with that option.
Besides that, sometimes it might be more comfortable to “bake” some “moves” into an animation clip and edit all of that in non-linear animation. (There—your Joint based idea would come in handy!)
A step that I would avoid would be point level animation PLA, as the tools discussed so far are way more powerful than PLA. But who knows, maybe that is the tool you like the most. You need to experiment with them.
All the best
The attached file is a sketch and not at all considered production ready!!