At the outbreak of war in 1775, most Americans were much more pissed at the British Parliament than they were at the king -- the rebellion was used as a means to get proper representation in the British Empire, not leave it altogether. George Washington and John Adams would even toast to the king's health during the first year of the conflict.
And their flag was also the one used by the British East India Company. The EIC controlled the trade in Britain's Asian colonies and was well-known to the founders -- after all, it was their tea that got ransacked by those pesky "Indians" in Boston. Before the outbreak of the war, American revolutionaries even considered the EIC an ally, as A) their trade was also being hurt by the British Parliament's taxes and B) the EIC represented an independent entity in the empire, something the colonists then aspired to be.
Which they expressed by playing the drum solo from "In-A-Gadda-Da-Vida" while dressed like a futuristic 1940s housewife.
Benjamin Franklin may have been responsible for the brief adoption of the East India Company flag. Sure, it enjoyed only fleeting popularity before it was (slightly) modified into the Stars and Stripes, but it definitely caught on better than Franklin's proclivity for a daily "air bath," a routine that required the stout Founding Father to sit entirely naked every morning next to an open window.