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.