There's no such thing as a literal interpretation of the Constitution
If there is a legal creed of American conservatism, it is originalism: the literal interpretation of the Constitution, according to the intentions of the people who wrote it. "The Constitution is a dead document," Antonin Scalia famously said. "It is an enduring document that does not change."
This idea is a ludicrous and impossible philosophy disguising narrow political self-interest.
Trying to make legal decisions based on the perfectly divined views of previous authors raises many immediate problems. Of most practical significance is changing circumstances. Any political document will inevitably run into unforeseen circumstances, and that is more true for the United States than it is for any other country. At 230 years old, the American Constitution is by far the oldest continually operating constitution in the world — the average constitutional lifespan is only around 17 years.
America is constantly facing developments the authors of the Constitution could not possibly have foreseen. It was largely written by aristocratic plantation farmers, many of them slave owners, who sharply restricted the franchise to property-holding men. It was a time before mass democracy, before mass production and industrial capitalism; before electricity and radio; before coal, oil, trains, planes, and the automobile; before the transistor, computing, and the internet. Even if we could divine the intentions of the authors with crystalline perfection, they likely provide limited at best guidance about how to make legal decisions about a modern wealthy state. It's senseless to try to predict what James Madison would think about, say, the proper regulation of internet platform monopolies.
But we can't see those intentions, because the (numerous) authors of the Constitution were often deliberately vague about them. When both the original document and many of the most important amendments were drafted was a time of intense political disputation, and often unclear language was used to paper over differences.
The 14th Amendment, for example, which has nearly constant ongoing political relevance (and has been the subject of innumerable court cases), was the product of a careful compromise between moderate and Radical Republicans over the status of freed slaves after the Civil War. Radicals wanted bold protection of black civil rights, especially the franchise, while moderates favored a complicated process by which states that disenfranchised their population would lose representation in Congress. The result was sweeping language protecting the "privileges or immunities" of citizens and their "equal protection under the law," as well as decreeing that they could not be deprived of "life, liberty, or property" without due process of law, but pasted together with the representation loss mechanism. Full black voting rights wouldn't come until the 15th Amendment.
But conservatives on the Supreme Court later advanced a hideously racist and extraordinarily strained interpretation of the amendment, saying, for instance, that it was unconstitutional to prosecute white terrorists who had butchered dozens of black Americans for violating their victims' civil rights, because it only applies to "state action" (incidentally, a phrase which does not appear in the amendment text). And when those same terrorists set up the Jim Crow caste system and disenfranchised states' black populations, the whole representation loss mechanism was completely ignored.
Finally, modern linguists and philosophers generally agree that it is not even possible in theory to remove all interpretation from textual analysis. When you read words — especially very old words — you must apply thoughts from your own mind to construct meaning from them. But you cannot access the mind of another person, especially one who has been dead for two centuries.
Of course, you can build an argument for certain passages meaning certain things. You can cite context, precedent, and history — and the case may be all but irrefutable. But in every case, you are doing interpretation, not divination.
In its ironically ahistorical lineage, originalism bears a marked similarity to Christian fundamentalism (as well as a similar political genealogy). The idea that the Bible could be used as a sort of empirical scientific document would have been barely comprehensible to the generations of people living at the time it was written, centuries before the scientific revolution introduced that set of concepts. Ancient Hebrew society was intensely symbolic and metaphorical, where everyday occurrences were presumed to have deep, non-obvious theological significance — not pinned down and stripped of all perceptual cognitive biases by modern empiricism.
This kind of pinched literalism comes from a childish lack of confidence in both instances. That's probably why in practice, originalism bears an extremely marked resemblance to "whatever the political needs of conservatives happen to be at this moment." Sometimes — as with the old Lochner doctrine, which made regulating the economy to benefit workers instead of business unconstitutional; or with the updated version of the same idea, which says unlimited corporate campaign contributions are free speech protected by the First Amendment but union contributions are a violation thereof — the level of tendentiousness borders on the hilarious.
Still, it's a pretty good slogan. Conservatives could simply try to argue moral and political principles and how they are consonant with America's legal traditions. But it's a lot easier to just point at their own particular interests and shout "Constitutional!" until they're blue in the face.