Right now, there is no time travel Genie in the new timeline. No one yet knows about the Guardian of Forever, no one knows about the slingshot maneuver outside of Spock. He has no ship and no capital to barter with outside of his knowledge of the future. And he's an old man.
Sounds like someone that could easily get taken advantage of if unscrupulous elements learned what he knows. You saw how easily Nero got the Red Matter from him.
I can come up with story reason after story reason why he would be reluctant to muck about in the timeline. It's not an white elephant.
Forget time travel. No sling-shotting. No Guardian of Forever. No temporal rifts to slip through.
He knows that the Doomsday Machine is coming and that it will kill billions as it gobbles up entire solar systems. His knowledge of how to destroy would not allow anyone to create super-bombs or to hijack the Doomsday Machine for their own purposes. He simply knows how to defeat it and save entire solar systems from being destroyed. Why wouldn't he share this information? Why shouldn't he share this information? Spock also know when and where plagues and famines will strike which will allow Starfleet to allocate provisions for these times of need. Kodos does not massacre 4,000 people if there is no food shortage on Tarsus IV. These are just two Star Trek episodes which should not happen and would not involve dangerously advanced knowledge being shared. We're just just talking useful knowledge. Spock can also warn Denova (and many other colonies that got killed before the booger monsters got to Denova) to protect themselves with UV light. This would simply save lives. UV light is not a state secret, even in our world.