Apologies - you're absolutely right, I've muddled the conversation by using the word 'azimuth' which is of course refers to a horizon measurement, which does indeed change with position. I was trying to refer the to lateral angular separation between the stars - the angle an observer would see if they measured it, or held up a suitable sight marked with angular graduations etc. That absolutely does stay the same regardless of position, which is how and why star almanacs can refer to star's positions by means of RA and dec number pairs - the numbers don't change with location, just their relative position in the sky. The time and date site has quite a good night sky simulator where you can change viewing location and see this effect yourself - I played with varying latitude in the northern hemisphere and it shows the principle very nicely - Polaris just moves higher up in the sky and all the other stars around it retain their relative positions - https://www.timeanddate.com/astronomy/night/@80,-0.
If you work out the angular separation between stars, they remain the same. That's not what your model would predict, as per my diagram - you would expect the angular separation to reduce as you got further away. It doesn't, though, does it?
The problem is that you want to use some parts of EA as if things operate in straight line geometry and think that you have identified a view and situation where it must apply. But this is incorrect. There is also distortion when viewing stars on a lateral view as well.
From this top down view an observer is observing two stars:
From a "3D" view of this below we can see that the closer star would create one angle, but if the curve of one star is dropping down to a lower elevation laterally then the rays of the second star would dip to a lower elevation and the angle the observer sees between the stars would not match the prediction of straight line geometry.
The points and curves I made here are somewhat arbitrary to show a point, but we can here that one star would create a greater curve than the other. The angle between the curved lines at the observer wouldn't make the same angle in space as the straight lines and angular separation as you envision it to be.
In another type of 3D scene where the light is shining from overhead and casts shadows straight down beneath a body, the following shows two lines (black) that appear at an angle. But if we look at the paths the shadows make on the surface, the shadows are actually intersecting at a broader angle on the surface:
You believe that if we turned the EA diagram I provided:
into a 3D view split into a three dimensional symmetrical cross insert, looking like a + from above, with four stars instead of two, that the observer at the far end of an arm would see two of the stars closer together than the top and bottom stars he observes. You think that an observer, who starts at the center of the scene beneath the stars, who then recedes to the end of one arm, would cause two of the four stars to get closer together and the effect would only apply in two dimensions.
But, as the central observer recedes away to the end of one arm the rays will dip in another dimension, causing them to widen out as they shrink.
It works in multiple dimensions, which is why the video I posted on the first page shows circles that are fairly circular.