No. It was the second-biggest port city in British North America (and would become the largest within a decade of the war's end), but colonial Americans* were distinctly aware that they were living on the periphery of a wider Atlantic empire, which itself often looked to Paris for fads and trends.

Check out Bernard Bailyn, The Peopling of British North America, for a conversation on the colonies as a periphery, not a frontier. Kariann Yokota, Un-Becoming British, looks at how Americans had to build confidence and a distinct identity during and after the American Revolution

*I know the song says it takes place in 1776, but the conversations the characters have in probably would have taken place in 1774. By 1776, Philip Schuyler was already a major general in the Continental Army, and I THINK Hamilton already had his command as an artillery captain. The conversation in '76 was whether or not the colonies should break away entirely, not over whether or not there would be a war. The war started in '75. As such, I'm still comfortable calling them colonial Americans at this point in the play, and chalk up the line about 1776 to Miranda wanting to anchor his play on the big years everyone already knew.