While I do agree that many Americans were fighting for independence more than for democracy, I think that it is a bit of an overstatement to say that democracy was only an afterthought.
When answering this question, it makes sense to remember that the American colonists were not one united and homogeneous mass. Different people had different motivations and different goals. There were some people who really did believe very strongly in democracy. Others really just wanted to be independent and to be the new rulers or an independent country.
It is clear from what happened after the Revolution that many of those fighting did not really care much about democracy. This can be seen, for example, in the way in which the country pulled back from democracy when it wrote the Constitution. The Constitution was meant to reduce the level of democracy in the new country.
On the other hand, there were clearly those who believed strongly in democracy. Thomas Paine was one such person. His pro-democracy writings were instrumental in getting many people to support the Revolution. Thomas Jefferson was also a democrat. The Declaration of Independence, which he wrote, is a strong statement of the need for democracy.
Because there were people like Jefferson and Paine who were active in the Revolution, I think saying the Americans only wanted independence is an overstatement.