- The transition might be visible as shown by video
What video?
The video you gave shows the sun with a sun dog. No transition.
- The ice crystals are scattered with relatively the same distant among those. This makes the sun light is reflected with different rates of distance of journey which consequently gives different light intensities. The more distant the sun light travels, the more absorbed by the air, and the less intensity at observers eyes. It occurs gradually based on the distance of sunlight journey.
That only applies to light which is passing through the air. For the majority of the journey between the sun and Earth there is no air to get in the way (or so little it is insignificant). So that doesn't explain why it can't happen with a distant sun.
In fact, the distant sun actually makes it a lot easier to explain.
When being refracted the light will typically refract at a specific angle, which is why sun dogs typically occur at specific angular separation from the sun, as do rainbows (including the higher order rainbows).
With a very distant sun, the light is basically coming in parallel, and thus will be refracted to the same angle. That means all the ice crystals along the path refract it towards you along the same line, allowing a bright sun dog.
With a close sun, those angles wouldn't line up, and you would get a smear instead.
So in fact, sun dogs require a distant sun, just like rainbows do.
In Saudi Arabia, for instance, when the temperature is higher than in Indonesia, but... we can feel that the heat in Saudi Arabia is more pleasurable than in Indonesia.
Consider the humidity.