What aspect of the European Parliament model were they trying to avoid by excluding this option?
The model used in the European Parliament hadn't been developed yet in 1789. The very first British government to fall on a vote of no confidence was Lord North's administration in 1782, due to the British defeat in the American Revolution. There, an explicit motion that the House had no confidence in the government caused North and his cabinet to be turned out of office; however, a new government was formed from current MP's, and the same Parliament kept sitting for two more years. So, the American Founding Fathers didn't provide a method for Congress to be turned out of office largely because none had been developed at the time.