Four Point Tracking doesn't do perspective

Hi. I’ve been watching this tutorial on four point tracking: Track, Match, Blend - 05 Four Point Tracking - YouTube
I tried this myself, and although I do get everything to work, I’m just running into one problem: the plane looks like it’s moving in 3D space, but in reality it still acts like a deformed plane.
This is especially noticeable when my plane is viewed at an angle. If it were true 3D, then it would have a perspective, but since it’s just a deformed mesh, it doesn’t. Or in other words, center lines in my mesh are seemingly sliding, they don’t stay centered, according to their perspective.

See this image:

To the left is a plane, viewed at an angle. To the right is a deformed plane, to look exactly like the one on the left.
However, it is clearly noticeable that the left one actually looks like a plane viewed at an angle while the right one has its subdivisions in ‘weird’ places. If you would measure this though, you’d see the subdivisions are actually at even distances, but that is not how perspective works.

Additionally, it is not possible to scale my plane just a bit larger or smaller than the area inbetween the four points.

So this four point tracking is not really working the way I’d want it to. I tried tracking my footage with more markers, so I could have the empties static, with a moving camera, but that doesn’t seem to work with just a flat surface. I don’t have any points outside my area to be tracked that I can use, I’m stuck with just a flat surface. And when I try to solve that camera motion, the trackers don’t want to be on a plane, they’re always at an awkward angle, and the camera is not at the right height either. Selecting three points and marking that as wall doesn’t do the trick either. It just rotates everything upside down and the awkward angles are still there.

Is there any way to do a four point track where the four points just stay put in a rectangle, while the camera moves?

It would be really nice if you could just say to Blender, “look, these four points are corners of a plane. Corner A is at exactly coordinate A (0, 0, 0 for example), corner B is at coordinate B and corner C and D are at exactly coordinate C and D, respectively. Now, position the camera in the Z-axis (using this FOV) so that that square perfectly matches these input points. Now do that for every frame.”

Basically, exactly how tracking already works, except with just 4 points on a 2 dimensional axis. Having to track an entire scene just to put up a 2D image is needlessly complicated and, as far as I got with it, more inaccurate than it should be. And simply copying the 4 points to empties and deforming a plane does not work with perspectives, as I’ve demonstrated.

You know that camera lenses are not perfect? They have distortion and may not match the dimensions given on the lens. So I am not surprised that it doesn’t work quite right.

I am actually tracking a cg scene, with ‘perfect’ lens. There’s no distortion, all lines are perfectly straight.
The problem is as I described, a deformed plane’s outline may look like it’s in 3D space, but when you subdivide that, you see that it is way off. The farther away side looks larger than the closer side, because the perspective is not properly applied.
Tracking the whole scene, as far as possible, having the part I wanted to track as a square (more or less) and having the camera motion does give the proper perspective, and then it doesn’t look wrong, because it’s mostly right. The four point technique may work for surfaces viewed dead on, or with a longer lens (so less perspective), but not with a wider lens at an angle.

Have you tried the BLAM addon? It infers reconstruction from shapes you trace.

Yes, that’s only for single images though? Not full motion tracking. I’m not really looking forward to do the same thing 960 times.
Haven’t really done anything with that yet, when I tried it, it wasn’t working too well, things still didn’t align good enough and the geometry reconstruction produced ‘wobbly’ meshes. And for some reason it thought the camera fov was somewhere like 14 mm, while in my own tests (with actual 3D reconstruction and precision camera placement), I came to the conclusion it was 20. Sliding a plane along an axis also showed it did not line up with the image.