If so, why haven't they started using Anglican after 200+ years and good relations with the UK?
The Church of England (or Anglican Church) in America was shattered by the American Revolution. Most of the clergy (who had to take an oath to support the King to be clergy) and many of the church members were loyalists during the revolution (For church members, this was more true in the Northern Colonies than the Southern, where many church members supported the revolution).
After the loyalists lost, most of the clergy left the country, as did many loyalist members. Anglicans remaining took a while to decide what to do. In 1784, they agreed to create a new constitution for the church in America, and revise the Book of Common Prayer for use in America (mainly taking out all those troublesome prayers for the English monarch).
They were troubled about how to create bishops and clergy for their new church in a way which would continue "apostolic succession". English bishops would not consecrate bishops without the oath of loyalty to the Monarch. They got around this by getting a Scottish Episcopal Bishop to appoint a bishop in America. (The Church of Scotland had a different arrangement with the Crown than the Church of England).
In 1789, an assembly of the American Church met in Philadelphia and voted to adopt the new constitution, a set of canon laws, and the revised Book of Common Prayer. In 1790, the few bishops the American church had managed to get consecrated began to consecrate others and appoint clergy, and the church began to rebuild, now separate from the Church of England. (Though very similar in doctrine and practice) with a different name, the Episcopal Church.
The American Episcopal Church, however was hit hard by the Revolution, and afterwards, never was as popular or represented as large a percent of the population as it had earlier.
Part of it, a mormengil mentioned, is that they take their name from their mother church in Scotland, which is called the Episcopal church ("Anglican" is still a touchy term there since the Episcopal Church was long persecuted and even banned). Another complicating factor is that since the 70s a number of more conservative breakaway denominations use "Anglican" in their names to indicate that they still have an affinity with the Anglican tradition but aren't Episcopalians.