I wonder if this is related to [Hyperfocal Distance](https://en.wikipedia.org/wiki/Hyperfocal_distance), a concept familiar to many photographers. Roughly, if you focus on the background (infinity), the foreground would be blurry; and vice-versa. If you focus about ~1/3rd into the scene, you'd have everything in reasonably sharp focus.
Unlike camera lenses, our eyes can't easily focus on an arbitrary distance without an object being present there. Perhaps the front sight is working as an approximation of the hyperfocal distance.
To stay hyperfocused, you would have to keep your sight focused on a certain dot (probably between sights and target), so it would be extremely hard to align iron sights without focusing on them as our eyes/brain automatically focus on things of interested. Peripheral vision training might help, but I believe, it still would be much harder than aiming in traditional way (focused on front sight only). Actually, our vision is extremely adaptive and after few sessions of practice shooting you don't even notice blurring anymore as it becomes natural.
"Perhaps the front sight is working as an approximation of the hyperfocal distance."
I haven't seen such gun yet. It would not be practical as distance of a target and illumination (two key factors for hyperfocus of human eye) varies greatly.
Unlike camera lenses, our eyes can't easily focus on an arbitrary distance without an object being present there. Perhaps the front sight is working as an approximation of the hyperfocal distance.