As our Republican compatriots love pointing out, we're a republic, not a democracy
Of course as a government teacher for the last ten years, I should clarify that these aren't mutually exclusive terms. The US is a republic and a democracy... and a federation of fifty states that are all to be guaranteed "a republican form of government" (Article IV, Section 4) by our Constitution. We're complicated like that.
I like to think about it this way: as a nation, as one people indivisible as a nationality, Americans are culturally democratic. That's the part of you that finds it sketchy that Colorado's Supreme voted 4-3 to remove Trump from the primary ballot. If we want to put Trumpism down, it won't be with legal technicalities. It can only happen with a strong political repudiation of his cult-like following at the polls in November. The law alone won't stop a movement rooted in believing it's above the law.
But as a country, as a legally constituted political entity called the United States, Americanism is in large part defined by a commitment to the rule of law--the law that we should all stand equal before. That's what we mean by guaranteeing a republican form of government. And the letter of the law is quite plain: someone who has taken an oath of office but then went on to engage in insurrection can hold federal or state office.
No person shall be a Senator or Representative in Congress, or elector of President and Vice-President, or hold any office, civil or military, under the United States, or under any State, who, having previously taken an oath, as a member of Congress, or as an officer of the United States, or as a member of any State legislature, or as an executive or judicial officer of any State, to support the Constitution of the United States, shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof.
Of course our Republican friends who no longer uphold the principles of either democracy or republicanism (or the fundamentals of truth) will cling to the idiotic argument that the presidency is not "an officer of the United States," and I fully expect Clarence Thomas and Sam Alito to cling to this fiction. They'll argue that the job of the president wasn't enumerated among the offices the rebs were disqualified from (Senator, Representative, elector), but as originalists they're also supposed to contend with the fact that the authors of the 14th never could have imagined any political party would be so foolish or so petty as to nominate a traitor to be president. Ha! Shows what they knew! Anyway, the argument that the president isn't an officer of the United States is moronic. Hence my expectations about Thomas and Alito.
But the grownup argument -- and this is what I've said from the beginning of this debate -- is that Donald Trump is so far accused of attempting to forcefully overthrow the government two weeks before his term ended, but hasn't been convicted of this specific charge. I think the evidence is overwhelming that he's guilty of it. But I admit I'm not unbiased. More importantly, there hasn't been a conviction by judge or jury on the charge--and I contend that because laws matter, because equal right for all matters, there needs to be a factual conclusion by a court of law reached proving that Trump rebelled against the United States. And maybe I'm being boneheaded in demanding legal certainty against the wild-eyed conjectures of Maga conspiracy theorists who don't even accede to the consensus reality. But where men fall for lies, we must always keep well lit the beacon of truth.
I am, as an American culturally, more eager to see Trump's threat to the nation put down by a small-d democratic vote, even if that's a risk that the "mobocracy" that the Founders feared could be hornswaggled by a demagogue into voting against their own liberty. And I am, as an American politically, convinced that the Coppertone Mussolini needs to be found legally guilty by a court before he can be disqualified under the laws of the Republic. The laws matter. Or at least, if civilization is on the line, the laws matter until they don't. Donald Trump is an agent of chaos. He threatens civilization. If civilization is to be held up, we have to uphold the laws that make us civilized.
In practical matters, of course, none of this matters. Regardless of how the SCOTUS rules on Trump's disqualification, he's only going to ever be removed from the ballot in a handful of Blue States where he couldn't have won the electors anyway. It's a hollow gesture to remove him from the ballot. It will have no effect, except to have validated his ridiculous argument that the system is rigged against him. As a partisan Democrat, I don't believe in validating ridiculous arguments.
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Jefferson tried to heal the bruised egos after the 1800 election by saying "We are all Federalists, we are all Republicans."
Later that decade, as the new Democratic-Republican Clubs began to form in the northern states, the still-informal name of the still-informal parties evolved to be called the Democratic-Republican party. But the longer name wasn't really in vogue until Madison's administration. During the Federalist era, most politicians would have recoiled at being called democratic. It implied chaos and street violence. And that impression deepened when the mobs of the French Revolution started to give democracy a bad name. After France calmed down and the natural logic of the US being a land of "equal opportunity" progressed, the word lost its negative connotations.
For that matter, since parties weren't formal structures, the name Federalist also began as a euphemism. During the ratification debates of 1787-88, the Federalists took on the name in order to get away from the accusation of being "nationalists" as they'd been accused of (implying they wanted to replace the 13 republics with one big nation-state -- if you read the original Virginia Plan, it wasn't an inaccurate accusation).
The word democratic didn't really come into popular acceptance till after the War of 1812, and particularly was embraced during the Jacksonian period. Tocqueville's observations on American democratic culture show a very different world than the transplanted English country culture that had dominated in the 1700s. Somehow the word Whig got politicized and evolved into the name of the anti-Tory, pro-liberty leaders of the late colonial period. A "whig" is merchant slang for an English countryside horseless cowboy, what we'd call a yokel. They worked as drovers and smelled like, as Monty Python would say, they were absolutely not kings. 
When it came time for the new anti-slavery party to organize on purer Jeffersonian principles, they took the name "Republicans" both because it was available and as an homage to Sally Hemmings's boyfriend. Once they started winning elections, the merchant class and robber barons moved in and quickly took over.
