So the average American sees their entire history as starting at 1776
Well yeah. That's kind of the way words work. Of course there was history before that with England. Which had history before them from France, German, Rome, etc. If we, US people, are talking about before 1776 with the colonies, that time is generally referred to as "Colonial History"
When the French stopped being a monarchy, it's gov't changed, the rules of law changed, it was effectively a different country. If a group of friends play football, then the next time they play basketball, they are playing different sports. Same people though.