If the shooting of President James A. Garfield happened in the present day, what would doctors do differently, and would he survive?

President James A. Garfield was killed by the infection caused by the doctors looking to extract the bullet from his body rather than the bullet itself. In fact, he lived for another 80 days after Charles J. Guiteau was said to have shot him in the back. According to Britannica, he even managed to sign an extradition paper from his bed.
At the time of his passing on September 19, 1881, doctors said his death was the result of massive haemorrhaging and slow blood poisoning. It if was today, he would most likely survive. Even if modern doctors were in a rush to take the bullet out of his body, they would do so methodically and in a sterile environment.
Vice President Chester A. Arthur succeeded him as president.
https://www.britannica.com/biography/James-A-Garfield