Sounds like you've got a problem with the management mind-set there, too, Chris. That's one of the weirdest "reasons" for a particular dress code that I've ever heard (and I've heard a few)!
The stethoscope has always been part of the medic's symbolism, of course. But (as I may have mentioned before), it's not too hard to appear smarter than (young) NHS doctors these days, casually dressed as most of them are.
Don't forget, guys (as if you could), that there is a whole world of biomed outside of the British NHS. In general, out there in the wider world, doctors still wear white coats, believe me.