Seeing as you've admitted that the size of the lights do not shrink linearly as they progress into the distance
Your claim is that the lights do not shrink
at all at a certain distance. Incorrect.
Of course the difference in size between a light at 10m and one at 20m is much easier to discern than the difference at 100m and 110m.
One is double the distance, the other is 10%.
So of course the further away something is, the less a small difference in distance will make. But we are not talking about the sun changing from 3000 miles away to 3100 miles.
That would be hard to discern (although it could certainly be measured with the right equipment).
We are talking about a difference between 3000 miles and ~9000 miles. That would mean an apparent size 3 times greater at midday as at sunset - maybe more, I don't know exactly what your estimate for the distance to the sun at sunset is.
So which is more likely? You're an Occam's Razor kinda' guy. What's the simplest explanation for a constant angular size. Is it:
1) The sun really is 3 or more times further away at sunset than it is at midday, but there's an unexplained optical effect which means that no matter the distance the sun maintains the same angular size 2) The sun is actually a consistent distance from us
PS: I made a bit of a hash of that image above. Here's another go. I've used a fill tool with the same sensitivity to fill in the last few lights in the stock image:
And then overlaid them
The change in size is clear.