In my technical drawing classes, I was taught that in three point perspective, lines that aren't parallel to the three vanishing axes are still straight, and you can in fact use that fact to construct additional vanishing points.
So for example, once you've established an axis-aligned cube, you can draw parallel lines across the diagonals of two opposite faces and they will meet at a new vanishing point, which you can use to construct more lines parallel to them, lying in that 45 degree plane.
It turns out that that vanishing point will lie on the line through the vanishing points that you used to define the cube axes, which is kind of interesting - each orientation of a plane in 3d space has a straight 'horizon' line in the drawing space, and parallel lines on those planes all vanish at a particular point on that horizon line.
I was specifically taught the diagonal projection technique particularly to transfer measurements throughout the perspective space - once you've established a diagonal baseline you can use it to transfer a length from one axis to another, which is great for doing things like counting out distances along an architectural facade to space windows. Laying out perspective-correct squares also turns out to be important for being able to correctly size ellipses for perspective-correct circles.
So I'm not sure where you got the idea that three-point-perspective as usually taught is a weird curved geometry. It's meant to be straight-line-preserving, which is precisely what makes it good for technical and architectural illustration. There's no such rule as 'don't get too close to your vanishing points' - it works just fine.