I second the, "time consuming and a bit stressful." ALWAYS time consuming, sometimes (oftn times) stressful.
FYI, I have slept in a car, with an unhappy wife (but at least no child at that time), when we couldn't find a place to stay some years ago in the Dolomites (not Swiss, I know, but similar situation). It was not a good night ;) We are better about it now. Smartphones have made "looking around" much easier but there is nothing like a reservation to take a load off of your shoulders toward the end of a long day. We have a couple of places around Switzerland that we go to if this sort of situation (ie no room available, no reservations) comes up. A place in Lucerne, Appenzell, near Saas Fee, outside of Lausanne.
We have found that reservations help to keep our marriage together.
One nightmare: We came out of the mountains around 7:30 (already pretty late by Swiss hotel standards). We stowed our gear in the car in the parking lot, I cleaned up our daughter, and my wife went up to the Alpine Gasthaus/resturant at the trailhead-- a pretty big place where we had eaten lunch the day before. At 8:00 they told her that we had a room (there couldn't have been but three cars there anyway). She came down and told us, we threw some stuff in a bag, got our daughter's things, and headed up the parking lot to the Gasthaus. We got there and suddenly the lady was no longer nice and it was, "too late". 8:11 at the latest--it had taken us a few minutes to pack some things for the night. I tried to talk to her and got, "no, too late now". I explained that we had a small child and her response shouted back was, "SHE'S NOT MINE!" I had to leave as she was escalating the whole situation. We drove back into the valley and stayed at a great little place run by a family that we had stayed at before. Great dinner and a great breakfast. BAD one: Berggasthaus Dammagletscher (I have NEVER had someone be so rude and angry--I just didn't understand what the issue was, really). GREAT one: Gasthaus Göscheneralp