Admiral Archers geliebten Beagle...

When talking about multiple realities, no matter whether the plot device that brings them about is time travel or wormholes or whatever, it is essential to clarify the fundamental concepts involved. Yet, people hardly ever do this, which is why many of the developing discussions are completely pointless. As a physicist, I can say that this is true for some of my colleagues as well - I've read more than one serious research paper that never even attempts to spell out the premises used, making any conclusions utterly worthless.

So, these are the fundamentals that I think one needs to consider:

Firstly, like most words in common usage, "existence" and "causality" have no strict definitions. Unlike many words in common usage, these are ultimately philosophical concepts, so a strict definition is a very hard to come by. Why do we usually understand each other when we use them, even if there is no strict definition? Obviously, because their meaning is plain in everyday circumstances. Common sense works well in understanding these circumstances, because common sense has been shaped by (in a Darwinian view) or for (in a Creationist view) precisely these circumstances. However, it directly follows that common sense is likely to fail when circumstances drastically exceed our common experiences. Hence, to continue using the same vocabulary, we would now need to fall back on agreed-upon strict definitions. Which, for the very concepts we'd like to talk about, we don't have.
With this background, it should be obvious that questions like "does alternate timeline A exist" or "does causality loop B give rise to a paradox" cannot even be discussed in a meaningful way until one explains what, in this situation, one means by existence and causality.

Secondly, fictional realities are tricky to start with, even in common-sense situations. Is a story that ends with the protagonist's waking up and realizing it was all just a dream less real than one that ends with them living happily ever after? Intuitively, most of us think so and find the former case somehow less satisfying than the latter. Yet, if one considers it rationally, there is really very little difference between the two cases. What one needs to realize is that in the former story, the protagonist and the universe that we have become emotionally attached to is the dream-protagonist and the dream-universe, not the dreamer and the universe containing the bed he sleeps in. The dream-universe is created just as directly by the writer as the "real" universe in the latter type of story. The distinction that is actually important concerns the basic rules of the two types of universes. Dream-universes are more flexible here than others - for example, they can suddenly end, when the dreamer wakens. "Real" universes don't do that. Unless, that is, when something like time travel is introduced, which brings us back to the point of this post.

Now, to make things a little less abstract, let's apply these considerations to a staple Trek time travel situation. The narrative flow follows the same three-act structure as conventional episodes: First, the situation is set up by somehow modifying the past, with negative consequences for the present to give the protagonists a motivation for their actions. Then, there are obstacles to overcome in attempting to repair the past. Finally, the situation is resolved by restoring the future to what it would have been without the incident.
Now, at this point, we (and, as far as we can tell, the characters) generally think of the alternate timeline as destroyed, simply because our point of view is returned to the original one. But, if you think about it, this is somewhat equivalent to claiming that the falling tree doesn't make a sound if there's no one there to hear it. The only thing we really know is that the writers chose to place the narrative focus on the restored timeline, and no more than that. And, applying the first fundamental problem outlined above, the question whether the alternate timeline still "exists" or not would be a tricky one even if this weren't fiction.

And, to apply the second one as well, are events that occurred in a destroyed timeline "less real" in any meaningful way than those that occurred in the main timeline? Only in as far as the former are not part of the characters' memories and don't influence the main timeline's future developments, while the latter are. And even this only applies to reality from the characters' point of view, not from the viewers'. For us, who aren't tied to any one of the story timelines, one is just as fictional as the other.

With all this in mind, it should be obvious that discussing a question like "does the original Trek timeline still exist at the end of the new movie" is something of an exercise in futility. The real question is this: Do the "old" and the "new" timeline constitute alternate realities in a single continuity, which we know as the Trekverse, or are they separate continuities based on different premises (Vulcan is destroyed or not). In the first case, crossovers are possible. In the second, they aren't. That's all, folks, thanks for reading.

ps: I do think that this approach is sound, and hard to disagree with on a rational level. However, it is of course perfectly fair to dislike it and continue discussing this topic based on a more intuitive take on these concepts. What one should bear in mind, either way, is that the intuitive take may differ between people, giving rise to disagreements which can't ever be resolved until one returns to a common, rational perspective.