Those vast expanses, those galaxies and suns, and that fragile Earth all featured prominently in Neil deGrasse Tyson's scientific story. I do not recall Tyson specifically referring to the mediocrity principle, but it buttresses his cosmology. Earth is unimportant, unremarkable; there are held to be uncountable earths populated with innumerable races. There must be: probability dictates it.
But the story of earth is anecdote, not data. We don't know how uncommon earth-like planets are, or how commonly they evolve some form of life, or how often that life evolves toward creatures like unto ourselves. Indeed, more recently the oddness of earth within the solar system has been heightened by suspicion that the formation and existence of our rather-larger-than-typical moon is important to the development of life here. The hope of those who want to believe that there is nothing remarkable about our human existence is that the unimaginably numbers of galaxies and stars within galaxies and planets around stars are sufficient to overcome any conceivable rarity of our situation, but pitting the unimaginable against the inconceivable is on the order of multiplying zero times infinity and expecting to get an answer.
The motivation behind this is Genesis, Chapter One, or rather, a distaste for it. When Stephen Hawking asserts that "the human race is just a chemical scum on a moderate-sized planet, orbiting around a very average star in the outer suburb of one among a hundred billion galaxies," the several value judgements (including Hawking's exaggeration of how common the sun is) are, after all, his judgements. God was not in the wind, nor in the earthquake, nor in the fire, so there is no reason to look for him in the gas giant, the supernova, the galactic core, or the black hole, or in any number of cosmic vastnesses or exotica.
If it offends that the creator of all should have created all but the most infinitesimal portion as mere backdrop to the divine earthly drama, well, that reflects on our aesthetics, not on God. The "pale blue dot" is a question of perspective, but in the end, it is the divine eye that matters.