Lesson in hospital visits

When I arrived home yesterday evening, I saw Sharon standing in front of her door with a giant bandage on her hand and a look of amused anxiety on her face.
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"So..."
"Sharon! Is that a bandage? What did you do to your finger?!"
"It hurts a lot. I think I need to go to the hospital."
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We rushed to put on our coats and left for CHUV, the university hospital in Lausanne. After running into Pasha on our way to Flon, we took the M2 to CHUV, which has the characteristic hacking and baby noises over the intercom to signify the stop. How clever. At any rate, we arrive to the emergency room and stand behind the yellow line until the receptionist is done on the computer. Sharon holds up her hand, and the woman gives us an apologetic look and starts to say something that doesn't sound promising. She calls another woman over, who proceeds to tell us that we have to go to a specialist clinic for hands and feet, called la Clinique de Londeraie.
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My question is, what happens if you are carrying your dismembered foot around and no longer have the ability to walk? Are they still going to make you go all the way across town to the specialist because the public university hospital cannot treat you? This is another one of those Swiss administration policies that I fail to understand. The receptionist printed out the instructions for us and explained the metro journey and the 15 minute walk from the train station, including which turns to take at which lights, etc. Sharon and I wondered why they turned us away if the hospital was completely empty, and we decided that it was empty because they turned everyone else away as well. We begrudgingly took the trek to the hand and foot clinic, which was on a small, sparsely-lit street. She ended up getting two stitches and was out in about 30 minutes. All in all, it turned out to be an important learning experience. Now we know that it is important to verify the correct hospital for your condition before you show up dying at the counter of the emergency room.