You just can’t be too clean in those sorts of situations. Also, they have to touch the gloves to put them on so it’s better to touch them with clean hands.
The idea is to make things as clean as possible to reduce the chance of infection. When dong surgery, it is not uncommon for gloves to be torn by an accidental needle, rough surface (e.g. cut bone), etc. Usually this is noticed, but not always. Why take a chance when washing has been shown to reduce infection rates?
A friend of mine who is now a qualified M.D. claims that the most difficult thing he encountered in his undergraduate years was learning how to put on sterile gloves without touching the outside…
In surgery, sterility is the key condition and while their gloves are sterile, there is always the possibility that they can be penetrated. All surgeons wash for a period of 5 minutes with an iodine based liquid soap, using sterile picks to even clean under their finger nails. The 5 minutes is to make sure that the soap not only removes all surface bacteria but kills bacterial lodged in crevasses of the skin, like around the nails, under the nails and then coats the skin with a temporary antibacterial coating.
They never touch the faucets with their hands once they begin the scrub, using foot controls to turn off the water or elbows. They dry their hands on sterile towels given to them by nurses who are already gloved and gowned.
The gloves, these days, act to protect the doctor as much as the patient with the amount of infectious diseases about. Should they break a glove, the potential for contaminating the sterile field and infecting the patient with germs is reduced almost to nothing.