Ok, but really wearing white clothes by no mean allows to better fix broken limbs. It just does not make any sense.
It'd be like "there are some scalpels and shit in pockets" and then it'd be weird that you can't just pickup those and use some normal suit or armor.
Because it's an equipment set. Doctor coat is white to make sure you know where to clean off the blood, and that if there is any blood spurting you it's easily recognized whereas darker clothes may conceal it. And the pockets and design allows easy maneuver and allocation of the tools, which assumably are included with the clothing.
By your reasoning it would require we have a separate item for each and every tool?
Unlikely possibility, yet this doctor coat is perfectly reasonable, as it is also already ingame.
(although, it doesn't contain all the sprites so meh)