Here the problem. Harry is aged at whatever he was when he went back. The older he is when he travels the worse the discrepancy. So present Harry couldn't go back and substitute himself say before Blood Rites before his his hand is burned. Or in Proven Guilty where Lash is still on board. Or where Molly exists. And he has the mantle. He isn't that younger Harry and can't be.
Or those things didn't happen, what if it is important that they did happen? If he makes them happen they may not have the same results that they did in the current timeline.. And may I point out not all the results of the negative things that happened to Harry in the current timeline were negative. Example, yes, he burned his hand severely in this timeline, but he also learned from it, that his shields had an important weakness that he was able to fix. He may be able to set it up so he gets burned again in the past, but can it be guaranteed that the outcome would be the same? The answer is, no.
Killing other versions of himself appears to be the plot device of Mirror Mirror.
Which brings us back to the original Trek episode, or rather a couple of them... Yes, in the original Mirror Mirror, Kirk tells the Spock of the other dimension/universe to assassinate the evil Kirk of that universe/dimension in typical Kirk fashion breaking the rules trusting that his judgement is right, but what if it isn't? Huge plot hole. In the other version again because of a transporter accident Kirk is split into his to personalities one overtly mild and moral, the other rather ruthless and aggressive, okay let's get rid of the "evil" or aggressive and ruthless Kirk.. Problem, to be a strong leader Kirk needs both sides of himself to be the leader he is, what is more without one side or the other he begins to die.
That is where I think the time travel episode, "The City on the Edge of Forever," is the best.. The lesson there was trying to do a good thing may be the worst thing for the future. McCoy saved Edith, good thing, she did a lot of good things for the world.. Problem, those good things prevented the world from stopping an evil in time, hence no human race, at least with the future of the Trek timeline.. So Kirk and Spock go back to restore the tragedy of the past, Edith, ahead of her time in many ways, had to die for mankind to progress.
Yesterday's Enterprise was another excellent time travel episode, only to be mucked up in later episodes by writers who don't know enough to leave perfect alone... Working that one out can make your head burst..