How do you think records get created? Someone has to do the initial research. It's not like archaeologists can just look everything up on Wikipedia (or the Star Trek equivalent). Someone has to have done the original research.
And then somebody would need to lose that research between 1962 and "today" (2260-whatever). It's not as if Kirk and pals could add anything of note to whatever the contemporaries already wrote about 1962, considering they are dilettantes when it comes to history, and are just visiting anyway, unable to grab or comprehend any context.
So, did Earth lose its history in a post-20th-century disaster? As per "Space Seed", interplanetary shipping records were a bit jumbled for 1996, but whether due to later loss or contemporary laxness in wartime bookkeeping, we don't know.
Why? There's nothing special about the Sun that other stars don't have.
When in doubt, "subspace". Sol could be different in many ways, some of them responsible for the fact that slingshotting only ever happens around Sol, others for the fact that heroes and villains in extreme hurry sometimes drop out of warp at Sol and proceed at impulse, even when everybody from Cochrane and Archer on jumps straight to warp from Earth orbit at other times without any particular hurry in evidence.
I wouldn't want to try slingshotting around really large stars or stars on the verge of supernova. That could get really messy.
And nobody in Trek does. But tellingly, nobody in Trek does it around nice and dull K stars, either. Except for Sol.
I read an essay many years ago in one of the "Best of Trek" books that speculates about the "disappearing bum" in "City on the Edge of Forever". If he hadn't accidentally vaporized himself with McCoy's phaser, history might have been completely different.
Sometimes our heroes or their bosses balk at altered history. But generally, they can't tell the difference (nobody we know ever
failed to have that bum vaporized, say). It's a bit difficult to see what sort of a "blunder" or "alteration" would make Starfleet distrust time travel, then. (Or Klingons or Romulans, who must have spies everywhere and be aware of slingshotting. Although if it can only be done around Sol, then they're screwed.)
Timo Saloniemi