Actually, whether the time travel of Kelvin-verse is different from the time travel rules of Star Trek is difficult to answer - since Abrams Trek (2009) does not contain any useful information about it. We see that there is some time travel and we see Original-Spock remembering a different past to the one he founds here. That's basically everything that we really know. There was a writers' strike in Hollywood when the movie was written and it is more as clearly showing when watching the movie.Well, the Kelvin-verse breaks/changes the rules of Trek time travel has had been firmly established across a couple of dozen or more episodes through the years.
So, most of the information about what the "Kelvin-verse" is (including its very name), is from interviews with the creators of the movie - in these they are buzz-wording some things like "alternate timeline", "parallel universe" and so on, but it only makes clear what the script of the movie already shows: that they don't have a clear understanding of what they did there. It's the same problem we see in the movie when they are referring to a wormhole with the term "black hole" or to Klingon battlecruisers with the term "Warbird". The only sound explanation is that they just buzzwording any term that they think could somehow be connected to the broader topic. So since they obviously can't use the right terms for the things they mention, there is no way to identify what they actually mean, if they say "alternate/parallel universe/timeline".
Long story, short: If they would have had any clear idea about what they are doing there, they would have given that explanation in the movie. It isn't there, so who knows what it is. But I definitely agree with you that it's a complete mystery why no one in the movie tries to undo the damage to the timeline.
== Time travel in Star Trek ==
For some different "types" of time travel that we can see in Star Trek, here is a first approach to categorize them. Actually, most of these categories are compatible to each other, so that time travel in Star Trek is actually pretty consistent. Note that the categorization sometimes depends on the perspective (e.g. "Accession" could also be Category 2F, if we consider Akorem Laan's unfinished poems as an already changed timeline).
Category 0: There is only one timeline and the time travel does not seem to have any effect (due to insignificant changes?)
- Category 0P: Someone travels into the past and back home; noone recognizes any (significant) changes to the timeline. (e.g. "All Our Yesterdays", The Voyage Home(the Bounty), "Wrongs Darker Than Death or Night", "The Sound of Her Voice", "Eye of the Needle", "Death Wish", maybe "Carpenter Street")
- Category 0F: Someone travels into the future; noone recognizes any (significant) changes to the timeline. (e.g. "Cause and Effect", The Voyage Home(Dr. Taylor, Whales), "Azati Prime", "What's Past Is Prologue")
Category 1: There is only one timeline all along, but it only exists because of time travel-induced cause-effect-cause-loop.
- Category 1: Someone travels into the past (e.g. because of some suspicious anomaly they found in the future), there they realize that they have to cause the events to happen exactly the way the original timeline remembers (e.g. "Assignment: Earth", "Time's Arrow", "Little Green Men", "Future's End", maybe "Captain's Holiday")
Category 2: A timeline gets changed to another timeline by time incursion and (mostly?) back again.
- Category 2P: Someone travels into the past and causes a change. By this, the original timeline is changed; but some people can remember the old version of the timeline; they undo the change as best as they can (e.g. "The City on the Edge of Forever", First Contact, "Past Tense", "Trials and Tribble-ations", "Relativity", "Storm Front")
- Category 2F: Someone travels into the future. By this, the original timeline is changed; the traveler is send back and the original timeline is (mostly) restored (e.g. "Yesterday's Enterprise", "Time's Orphan")
Category 3: A timeline gets changed to another timeline by time incursion.
- Category 3P: Someone travels into the past and causes changes to prevent some events from happening (e.g. "Timescape", Generations, "Visionary", "The Visitor", "Children of Time", "Endgame")
- Category 3PE: Someone travels into the past and causes changes. By this, the original timeline gets eliminated and the time traveler fades from existence. (e.g. "Time Squared", "Parallax", "Timeless")
- Category 3F: Someone travels into the future. The traveler is send back and the original timeline is changed (e.g. "Accession")
- Category 3I: Someone sees a possible future and with this knowledge the timeline gets changed (e.g. "All Good Things...")
- Category H: Time hick-ups (e.g. "We'll Always Have Paris", "Future Tense")
- Category I: Someone travels in time, but it is only an illusion. (e.g. most likely "Tapestry", maybe "Terra Firma")
- Category U: There is some time travel. We don't have enough information about the original timeline to make any guess about changes. (e.g. "A Matter of Time" (?), "Firstborn"(0P or 1))
- Category U2: There is some time travel happening in a greater time travel context. Due to the complex context it's difficult to say what the "original" time is or what changed (e.g. "Cold Front", "Shockwave", "Zero Hour")
- Category R: There is some time travel. I don't remember the details well enough. (e.g. "Before and After", "Fury", "E²")
- Category ?: There is just a lot of time travel. (e.g. "Shattered")
- Category Red Hering: The "story-device" "time travel" is used to explain away plot holes without giving any reasonable explanation or any motivation of the character's acts. (e.g. "The Red Angel", "Perpetual Infinity", "Such Sweet Sorrow", "That Hope Is You, Part 1")
See also: List of time travel episodes at Memory Alpha: https://memory-alpha.fandom.com/wiki/Time_travel#Background_information
Live long and prosper,
Norad