This is an interesting question. I don't know of any scholarship on Sun imagery in the Early Modern period (I could imagine a nice research paper coming out of someone willing to parse over a lot of these things).

It is of note that all of those examples from the 17th century are heliocentric depictions. Early advocates of heliocentrism, especially Kepler but also others, did often imbue their arguments with aspects of Sun-worship, and waxed poetically in anthropomorphic manner about the Sun. Here is Copernicus, from De Revolutionibus:

In the midst assuredly dwells the Sun. For in this most beautiful temple who would place this luminary in any other or better position from which he can illuminate the whole at once? Indeed, some rightly call Him the Light of the World, others, the Mind or the ruler of the Universe: Hermes Trismegistus names him the visible God, Sophocles' Electra calls him the all-seeing. So indeed the Sun remains, as if in his kingly dominion, governing the family of Heavenly bodies which circles around him. So the Sun sits as upon a royal throne ruling his children, the planets, which circle round him. ... The Earth has intercourse with the Sun, and is impregnated for its yearly parturition.

The notion of the Sun as a God, and notably the Sun as a king, is very prominent in this kind of imagery. Here is Kepler's argument:

[The sun] is a fountain of light, rich in fruitful heat, most fair, limpid, and pure to the sight, the source of vision, portrayer of all colors, though himself empty of color, called king of the planets for his motion, heart of the world for his power, its eye for his beauty, and which alone we should judge worthy of the Most High God, should he be pleased with a material domicile and choose a place in which to dwell with the blessed angels.…

For if the Germans elect him as Caesar who has most power in the whole empire, who would hesitate to confer the votes of the celestial motions on him who already has been administering all other movements and changes by the benefit of the light which is entirely his possession? ...

[Hence] by the highest right we return to the sun, who alone appears, by virtue of his dignity and power, suited for this motive duty and worthy to become the home of God himself…

And so on and so on. And of course we have rulers who associate themselves with the Sun as well — Louis XIV, famously "The Sun King."

So the imagery of the Sun as some kind of king of the skies, looking on the Earth with his light, and maybe even (as Kepler would hold it) being the seat of God himself... this is very common in this period in European writing about the Sun, and extremely common in early heliocentrism, which itself had many explicit references to Ancient sun-worship (e.g., the Pythagoreans). It is largely what drove men like Copernicus and Kepler to be heliocentrists in the first place (only later did they get some evidence for it, and even then it was somewhat ambiguous), in the same way that those who advocated for geocentrism were also driven by aesthetic, religious, or philosophical motivations. But I would note that to my knowledge neither Copernicus nor Kepler ever depicted the Sun in such a grandiose fashion, visually: their hyperbole was strictly verbal.

Anyway, that's my guess: a mixture of reverent anthropomorphic imagery with a visual stylization ("Sun has a face") that — accurately — reflects the prominence of the Sun in the Copernican worldview (as opposed to the Ptolemaic one, in which the Sun is but another planet).