This is not surprising. Many colonists considered themselves loyal subjects of Britain -- many colonists came from Britain, and King George III ruled over the colonies.
In Boston, on that New Year's Day, the Loyalists (supporters of Britain) had been circulating a recent King George speech, offering the Continental forces favorable terms if they laid down their arms. These Loyalists were convinced that the King's speech had impressed the Continentals into surrendering -- as a sign of the Continentals' "surrender," the Loyalists mistook the flying of the Grand Union flag over Prospect Hill as a show of respect to King George. In fact, however, the Continentals knew nothing of the speech until later. Washington wrote in a letter dated January 4, "By this time, I presume, they begin to think it strange we have not made a formal surrender of our lines."
Obviously a new flag was needed.
According to Betsy Ross's dates and sequence of events, in May the Congressional Committee called upon her at her shop. She finished the flag either in late May or early June 1776. In July, the Declaration of Independence was read aloud for the first time at Independence Hall. Amid celebration, the Liberty Bell tolled, heralding the birth of a new nation.
Much suffering and loss of life would result, however, before the United States would completely sever ties with Britain. Betsy Ross herself lost two husbands to the Revolutionary War. During the conflict the British appropriated her house to lodge soldiers. Through it all she managed to run her own upholstery business (which she continued operating for several decades after the war) and after the soldiers left, she wove cloth pouches which were used to hold gunpowder for the Continentals.
On June 14, 1777, the Continental Congress, seeking to promote national pride and unity, adopted the national flag. "Resolved: that the flag of the United States be thirteen stripes, alternate red and white; that the union be thirteen stars, white in a blue field, representing a new constellation."