Some things about altering the timeline:
1) Spock can't alter it too much to one party's advantage or others would do it too. I.e. Sure he can warn the Federation about the destruction of the Deneb colony, but he'd also have to warn the Klingons about the loss of their Ch'whatsit colony by the flying killer brains of Jabnab IV.
Then he has a moral obligation to do both. Share as much information as he can about avoidable tragedies.
2) He can't give the Feds too much in the way of weapons knowledge or the Klingons would have a fit and send their own emissaries back in time or creating alternate timelines/dimensions - possibly ones that would come to the aid of their brothers in this one.
This is the same Spock who was willing to break treaties and violate the neutral zone with Kirk to steal just one cloaking device. There's no need now as Spock can discretely disclose to the Federation how Klingon cloaks work. And that means that the Federation does not need to risk a war.
If Spock, as you say, should be concerned about the balance of power, his superior information would allow for a much less risky balancing of power.
3) If he told everyone about the loss of the E-C saving Narendra III, would the Klingons believe him securing peace earlier; would pro-war parties in their government make a war happen sooner not wanting peace with the Feds; would the Romulans strike early?
He's not obligated to give everyone all the details. He is, however, obligated to the mission of the United Federation of Planets. He's in a new timeline, so he can't be entirely sure that things will play out to the advantage of the Federation, so he should do what he can to make sure that delicate causal nodes in history don't tip the wrong way (e.g., the Enterprise Incident).
4) Even giving medical knowledge to everybody he could remember, what if the Tzenkethi got all pissy their diseases weren't remembered and started a war over it. Or if the Sheliak found the spreading such knowledge as dangerous as the proliferation of WMD and started a war over that?
This seems much less plausible. Medical knowledge would simply be welcomed. And the UFP doesn't have to tell everyone how they found it.
If certain medical knowledge is dangerous, he might hold that knowledge back.
All that said, I don't know I wouldn't spill everything I knew were I in the position to do so, hoping that the info I spread would be used for the best. Still, for the entertainment franchise that is Star Trek, it can't be changed so much that it's unrecognizable to its fans/profitable.
Which is why this is a nice little tension in our new universe. The very existence of old Spock in this universe means that someone who knows the spoilers details of the original 5 year mission (and much more!) of the Enterprise is there to tell them
*He's not a Greek God. Refuse to worship him and he'll lose his power.
*Give the Defiant a wide berth.
*Yes, it is vampire cloud monster. Here is how you kill it.
*The Gorn believe that these planets are theirs. They will totally kill anyone who tried to settle here.
*The horta is not your enemy.
But look at it from another angle! Old Spock is now a reason why they can have totally new adventures. He can warn them off of needless dangers so that they may face different challenges.