I'm using an extreme example to illustrate that you can't assume the gravity on an arbitrarily shaped body reacts similarly to that on a sphere. With a really flat sheet, as you pointed out, you have to look at a completely different direction. In my cuboid, which is somewhere between the sheet and a cube, you'd have something in the middle - so you really can't just use common sense.
Similarly, you can't assume that the pressure on an interior point is determined by a vector normal to the surface. That only works on a spherically symmetric object.
Last time I've taken static mechanics was in 10th grade, so it's been a while. It's great for calculating stress loads on bridges. It is NOT meant to be used on astronomical calculations. If you don't believe me, show your calculation to your teacher and see what he/she says. The odds are you will be pointed to a vector field methodology, since that's how it's been done since Laplace.
Too much whining, too few results. If you want to use Laplace, just do it! If you want to show us how to do it, just show us how you did it.
You still try to explain away the mountain of results that are already shown by arguing against the use of the third law of Newton, which is exactly what you do when you dismiss the simple discipline of Statics. But you are not even saying why you do not like it, just that it is learned in tenth grade. (Edit: and that it is meant for bridges, not for blocks; how on Earth the third law of Newton (sum of forces = zero in isolated systems) get banned from blocks if they are big?)
I would accept your argument that the complete vector field should be calculated if the pressures inside the block were close to the limit of pressure that the metal can withstand without permanent shape change, but I have demonstrated that the pressures are thousands of times those required to permanently change the shape of lead, and even hundreds of times that needed to bend steel. Even if you find that I made a big mistake and the pressure is one tenth the one I calculated, this is still thousands of times too much for the lead block.
What you have not brought up and that I have not explicitly calculated is the pressure on a plane orthogonal to the main axis. Of course, if you had done that part of the work you would have found that this pressure is a small fraction of the one exerted parallel to the main axis, so this would confirm that the block would not keep its shape.
So, please, show you are a mathematician and show some results, you know, those things expressed with numbers and names of units. Any precision better than one order of magnitude will be better than your present performance, which is nothing, nada, null, rien, мелочь.