I know this isn't going to win me any friends, but a Republic is a type of democracy. It's a Representative democracy. A democracy is a system of government where the people rule, as opposed to a Monarch or dictator or such.

A representative democracy, such as a Republic, places power in the hands of the representatives that the people elect. Our Founding Father's didn't want mob rule, an absolute democracy, where the people represented themselves. But they did want a government of "by the people, for the people" instead of a monarch and aristocracy.

There are many types of Democracies. I think people want to argue that if we aren't an absolute democracy, we aren't a democracy at all. But that's just not true.

Don't let your frustration with a political party override logic and basic word definitions. Would people be so against the idea that the US is a type of democracy if the Democrat party was called the Progressive party?

I've noticed that the rejection of the US being a Democracy has been a pretty recent thing in the Republican party. It really does seem to have come about in the Obama era.

For instance, here are some quotes from Reagan;

https://www.azquotes.com/author/12140-Ronald_Reagan/tag/democracy

This democracy of ours, which sometimes we've treated so lightly, is more than ever a comfortable cloak, so let us not tear it asunder, for no man knows, once it is destroyed, where or when he will find its protective warmth again.

Ronald Reagan

America is a melting pot, and education has been a mainspring for our democracy and freedom, a means of providing gifts of knowledge and opportunity to all citizens, no matter how humble their background, so they could climb higher, help build the American dream, and leave a better life for those who follow.

Ronald Reagan

In America, our origins matter less than our destination, and that is what democracy is all about.

Ronald Reagan