I disagree vehemently.
nuBSG was alive, dangerous, unpredictable, AND most important of all: socially relevant.
And it's that "socially relevant" quality combined with all its other factors that made me think of Enterprise and what it should've been.
Given that Enterprise took place many many years prior to TOS, it's not inconceivable to me that it could be different and have a BSG tone.
There's different ways to go achieve the "alive, dangerous, unpredictable, socially relevant" aspects of storytelling, in sci-fi, legal dramas, police procedurals, dramedies, etc. Other fictions do it in their own styles, without resorting to BSG-type storytelling. TOS did, sure. But Trek is fundamentally brighter and more optimistic than BSG, with a far greater emphasis on exploration (space, time, personal, political, social) rather than survival, and therein lies the key difference.
Consider that DS9 is without a doubt the darkest of the Trek shows, and perhaps the most introspective of itself and the one that offered the most critique of the Trek utopia. It's no coincidence that the BSG writers cut their teeth on DS9. And yet, Trek's own nature wouldn't let them go with a BSG tone simply because there *was* that utopia that they were analyzing (otherwise, there'd be no point in that self-reflection). Trek tropes were there, while still maintaining its social relevance and other traits that you described; especially towards the end, it explored religion, occupation, fascism, terrorism, ends-justifying-the-means (all of which happens in the real world), and ultimately a rare peace between Trek's 3 most popular nations. And DS9 definitely felt more like Trek than BSG while still hitting those core 4 traits that you mention.
BSG is its own creature, and rightfully so; it's a show that's primarily concerned about politics, coping, and survival, and that's perfectly fine. Trek is its own creature as well, with a broader focus on exploration. Trek's failings come when it tries too hard to be other things. BSG and Trek can succeed on their own terms, but they can still achieve those traits you wanted while still retaining their unique identities, because those traits are only ingredients, not the entire product. One only need look at their respective finales -- BSG's ending had them make it to Earth and survive, as the culmination of a 5 year investment by the viewer, succeeding in their mission and surviving to create a new start. A few of Trek's finales, on the other hand, were about contemplating future and fate beyond survival, about life and growth after the hardest of challenges, the beginning of a new exploration. Neither BSG's nor Trek's finale premises are inherently better than the other, but rather it's about which ending is more fitting for each show's individual philosophy.
BSG is fine, but Trek really should just stick to being Trek -- and we've had several series where Trek was indeed "alive, dangerous, unpredictable, socially relevant" (granted, some series handled it better than others). Should Trek adapt to a 21st century style of storytelling the way many modern shows are doing? Sure, definitely, but it should also still retain its Trek identity as well. One need not be BSG to achieve those goals, as other programs have shown us.