I'd argue it's a tell that they don't understand the history of the term or the perception of it held by those who disagree with them.
For most people who say it, I believe "we're a republic, not a democracy" means "we're not majority rule, there are structures in place to protect the interests of the minority". This is why they don't trot out the phrase every time the word "democracy" is used. It's a rebuttal to specific instances where people invoke "democracy" to mean "most people I know disagree with you, so your opinion doesn't matter".
Maybe I'm projecting my own views here, but I generally agree with what I believe people are trying to convey when they use the phrase. I don't use it myself, because I know it's not an effective way to communicate my position - not because it's technically not a true statement.
For what it's worth, there's also more nuance to the word "democracy" than you seem to be presenting here. At the time of the founding, it basically meant "rule by popular vote", and was pretty much a synonym for "mob rule". A large part of the US system was explicitly designed to work around the shortcomings of direct democracy.