I know exactly what you mean. When we were in Switzerland over New Years, we visited a town called Rapperswil on a day when the majority of the country was enjoying a national holiday so all the shops were closed and there wasn't really anything going on. We practically had the town to ourselves! It was amazing.
Sounds like it was a perfect day to explore! Speaking of New Years, we were in Malta at the time and expected it to be deserted on January the 1st but the streets were everything but empty. Guess it depends on the country and the traditions...