I need everyone to understand that one of the greatest tragedies of McCoy's life is that he was born too late to use a stethoscope. This said, I like to headcanon that maybe his daughter gave him an antique stethoscope as a gift and he grew to love it with the force of a thousand suns, for three reasons I will now outline.
Reason number 1 - McCoy doesn't like technology.
Sure he'll use it, but he'd rather go for a more hands-on approach than rely on some machine whenever possible. A stethoscope would let him literally hear someone's heartbeat and breathing instead of just seeing readouts projected on a screen.
If someone uses a crazy space-EMP to knock out all the technology in medbay, you know what's not going out? A stethoscope, that's what. If they're going to a less-advanced planet and they can't bring their crazy future-tech, you know what they maybe could bring? A stethoscope.
Reason number 2 - while using a stethoscope, your patient isn't allowed to talk.
Imagine it. "Whoops, Spock, need to check your lungs again. Deep breathing!" "Oh, no, Jim: gotta listen to your heartbeat nopenopenope no arguing doctors orders!"
He would be unstoppable.
Reason number 3 - stethoscopes are infamous for being super cold.
This would give him another weapon in his arsenal to harmlessly harrass the patients in his medbay. He would spring the stethoscope on Jim without warning to make him jump, and when Spock is coming down to medbay he would dunk it in ice.
In conclusion, McCoy was born to wield a stethoscope alongside a hypospray, and clearly the only reason he does not do so is to prevent him from becoming too powerful.