In all other Star Trek shows and movies, time travel was shown as affecting the universe of the time traveller and not creating a new universe.
True. But MWI does not conflict with what happened there.
- Every Backward trip creates a new reality from that point onwards.
- Every forward trip moves forward within said new reality.
- Most of the Time Travel stories are told from the perspectives of the characters.
That is an assumption. As explained above, the depiction of Time Travel linearly does not actually conflict with each backward trip creating a new reality.
Given your assumption, this sounds logical.
In any event, unless the red matter and super nova altered the laws of (Star Trek) physics in the new universe, Spock' could have time travelled in methods used in previous incarnations of Star Trek (sling shot, Guardian) and travelled back in time to save Vulcan and alter the future of the new universe. Apparently Spock' chose not to do that.
If we assume that Spock is aware of the Alternate Reality, regardless of your assumption, Spock's sense of ethics would preclude interfering with the natural course of events.
Even if he found a way to get back in time to the Narada's arrival, he would need untold resources in order to stop the change.
He would also have to ensure that he can stop the Narada from emerging from the "lightning storm in space", which there is likely no physical capability to do.
If he destroys the Narada, the battle would be observed by the Kelvin, and the destruction of the Kelvin is still likely. One missile shot, and things may be even worse, with Kirk not being born at all.
If he tries to find a way to self-destruct the Kelvin, he would still be dealing with the butterfly effect from the anomaly and the Narada appearing, in front of the Kelvin, and exploding, leaving all kinds of advanced technology, which the Klingons would want to use, and would tip the balance of power in unpredictable ways.
Looking at the options, it would be difficult, if not impossible, to actually make such a change.
All of this reasoning and thinking, which Spock likely would have done, is simply too much technobable and explanation for the movie, and would not help the central story, but intsead would simply confuse the audience.
Therefore, the best way forward would be to simply leave it alone, which is what was done.