Isn’t it just trying to repair the timeline the same as what Kirk did in City On The Edge of Forever though?
No. Telling Cochrane about a giant statue built in his honor does not fix the timeline. Just because the original cover story didn’t make traction doesn’t mean they revert to telling the truth, the whole truth, and nothing but the truth—even bits that aren’t relevant to accomplishing the mission.
And if I recall, Kirk didn’t actually tell Edith Keeler (or anyone else) they are from the 23th century nor on a mission to see her death takes place. In DS9, when Sisko was confronted with fixing the timeline after Gabriel Bell was killed, he found a way to complete the mission to (mostly) restore the timeline without telling anyone the truth.