As for "things can drop on your feet", that rule would make sense if they also banned, say, tennis shoes or open sandals, which provide no protection for the top of your foot. (And if they didn't try to claim it was a health department thing. I hate misinformation about easily-verified thingies.)

Obviously, in a hazardous area, I'd wear some sturdy shoes, or not enter at all - I don't want to get my foot broken (or lacerated), I break my toes often enough as it is! And a broken foot is hard to heal. But my everyday shoes, and probably your everyday shoes, and most people's everyday shoes aren't likely to provide much protection from that sort of thing - and I do believe that in most situations, heavy shoes are harmful instead of helpful.