If we knew the exact location of 3 stars at one moment in time, we would be able to measure the angels between them and then use triangulation to work out the locations of different countries, and then make an accurate flat earth map.

See https://en.wikipedia.org/wiki/Triangulation_(surveying)
The principle would be the same as here, except working in 3 dimensions instead of 2, hence why we need 3 points instead of 2.

The formula for calculating your location takes the 3d coordinates of the 3 stars you are using, and the visual angle between each pair.

I thought I could look at star charts for the night sky in different countries, and use more stars than 3, I could use some simultaneous equations to calculate the exact location of the stars, using only the angles we can see in those countries, and without using any of the fake distances round earth maps have.

Here I found a problem. If you take 2 stars from two well known constellations, apparently the visual angle between them is the same no matter where you are. Now this makes no sense, because the triangulation formula would then mean that all these places are somehow 0 meters from each other. The difference in visual angle between countries can't be too small to measure, because that would mean that the stars are very far away, around 50 thousand miles away if you can't see a difference of a degree when you move a thousand miles away. It wouldn't make sense for a star that far away to appear high in the sky in one place, yet near the horizon in another.

What am I missing here? The results of the math seem to be nonsensical.