This timeline had already be altered 25 years before Spock Prime emerged. In effect, when he emerges, the damage has been done and Spock Prime is now a part of this new timeline. There is nothing he can set straight. He might as well be a part of it.
To that end, nudging Kirk in the direction of taking over the Enterprise is probably the point they are addressing before Kirk beams out. Kirk certainly wouldn't have done that without Spock Prime's urging and help. Who knows where he would've ended up had he not been told to take that chance by Spock Prime?
As far as Spock Prime's response to changing history being cheating is that he learned it from an old friend goes, the only problem I had is I don't recall a time Kirk ever set out to change
history. In TVH nothing was changed. Whales were brought to the 23rd century, but obviously that was supposed to happen. In "City" or "Tommorow is Yesterday", it was about returning things to as they were supposed
I think the thing the writers may have done is play on the popular misperception that Kirk made many trips through time and altered history as needed. Which is, of course, patently wrong in reality.