Someone had to run things in between.
Someone did run the government between 1781 and 1789...The Congress. If anything, these so called "Presidents" were merely a figurehead. They had no power. Which is one reason why we decided to abandon the Articles of Confederation. Hence, Washington really is the 1st President of the United States (unless we count the others based simply on what they were called).
The British surrendered in 1781. Washington was inaugurated in 1789.
Someone had to run things in between.