How did Tony fix himself by the end of Iron Man 3?

My theory goes with the assumption that He could have removed the shrapnel at any time after coming back to the States

Bear with me.

Imagine, you're Tony Stark. You're told by a genius scientist that he can't remove a piece of shrapnel due to sterilization and equipment purposes.

You become dependent upon the mini-arc reactor that you put inside your chest to power your escape. You then use this same reactor to become the most famous man and renowned "Super Hero" to the people, absolving you of years of weapon making and death that you yourself engineered and sold.

Would you want to remove that piece of yourself that reminds you everyday to keep being that crime-fighting badass that everyone loves. To keep being awesome and not fall back into old habits.

Because of this, it turned from a partially inoperable piece of shrapnel into a mental dependency on his mini arc reactor.

However, after dealing with his apparent PTSD from NY, dealing with trying to make things work with Pepper, and dealing with the tragedy that befell Happy. Combined with perfecting suits that are autonomous (do not need him to pilot or power them) he finally realizes that he does not need any of it to continue being who he is.

After Killian's apparent death, he was able to finally accept that he needs to move on, start hanging up his suit (his Clean Slate protocol), stop depending on the arc reactor, and start living his life in true happiness instead of fear of what happened in NY and instead of whatever apathy he was feeling being a playboy.

In essence, it is a symbolic representation of Stark's growth over the movies. Coming from another rich genius to someone dependent upon everyone loving him for being Iron Man, to now just wanting Pepper to love him for who he is.

During the scene where Tony has the surgery, you can see a miniature ARC reactor and some gadgets above Tony. As the surgeon removes a piece of shrapnel, he releases it and it flies up to the gadget. So Tony and the surgeon built an electromagnetic device to help out with the surgery. The eagle-eyed will also notice that when they show Pepper and Rhodey watching the surgery, the signs on the doors in front of them are in Chinese. Shortly afterwards, Tony gives Pepper a necklace, and behind them is Shanghai's skyline. He went to China for his surgery.

During the credits, you'll see there were several movie studios involved in making the film. Marvel Studios and Paramount are familiar names to people who watch Hollywood films. DMG Entertainment was another one, which may be more familiar to fans of Chinese cinema. They're a Chinese company that in recent years has been investing in Hollywood movies. They do this, and Hollywood movie studios agree to this, because by working with a Chinese company, the film then isn't treated like a foreign film. There are limits on the number of foreign films that can be released each year in China, this sidesteps that. In return though, DMG has extra China-centric scenes filmed that are then included in a China-only cut of the film (this happened for both Looper and Iron Man 3).

For Iron Man 3, the China-only scenes explain why Tony went to China for his surgery. At the start of the film, in Switzerland, Tony briefly meets Ho Yinsen and Dr. Wu, a cardiologist. Dr. Wu is the one who performs his surgery. In the non-China cut of the film, this isn't really explained. During the China-only scenes, it's stated that Dr. Wu is the only one who can remove the shrapnel from Tony's chest. So Tony goes to China and gets the shrapnel removed, which is something he couldn't do before.

Tony cured Pepper. Tony perfected Extremis to regrow his breast bone and ribs which were cut open in a circle for insertion of the electromagnet/arc reactor. So, he needed a perfected or reversible Extremis to perform the surgery because of the missing bones and then to be cured of Extremis afterward. I believe without Extremis that he would die from the surgery. He said something to the effect since he was able to cure Pepper, why not do some cosmetic or minor surgery. I am sorry not to have the exact quote.