Of course they should (wash them, but not necessarily IN the room)! I wash mine in "the back" before going in a room, even though there are sinks in the exam rooms. Somebody asked me to wash my hands in front of them recently, but I find it awkward, like I'm being watched, and rushed. So now I finish wiping my hands w/the paper towel as I walk into the room and throw it away in the exam room trash. It's so silly how much effort we put into appearances rather than what's really important.
And yes there is an excuse for not using the water-free sanitizers. For me, it's ECZEMA. Those things STING!
We also have this "liquid glove" lotion that acts as a shield against the buggers... even mites! Very cool, and it's one of the few lotions I've used that moisturizes, isn't greasy, and doesn't burn.