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The problems with the tariffs-dunnit argument are numerous and you're only really asking about the documents in question, but I want to go over them to lay the groundwork before digging in because the idea is just so common. Some of these are pointed out in the WaPo article you link, but here they are with a few more:

The "It was Tariffs" people ask us to believe at least that

1. The South hated tariffs so much that it seceded to escape them after writing tariffs for decades and consistently negotiating lower and lower rates

2. The South would somehow have no ability to block or negotiate favorable duties with a new protectionist party despite having done so with the previous one and there being no fundamental change in how bills became law.

3. The South was unified in loathing tariffs, despite the fact that an openly protectionist party, the Whigs, was at least serious second party and occasionally dominant in some southern states for decades prior. Sugar planters in particular depended on a tariff to protect their own high prices against foreign imports. Louisiana, where most of them lived, seceded.

4. By far most tariff duties actually paid were paid in Northern ports, the vast majority in New York, not out of Southern pockets.

5. That the tariff harms agriculture, the main occupation of most of the country, but for some reason all the Yankee farmers don't see it as secession worthy but most of the enslaver farmers do and this is just a coincidence.

6. The previous big tariff fight that threatened disunion was much more explicitly about a specific and very high duty. South Carolina challenged that law to the breaking point...and none of the other states came along for the ride. Furthermore the political leaders of that movement included men who had growing profits in their ledgers among its more radical leaders and those leaders admitted that the tariff was simply an alias for slavery intended to make them appear more high-minded. This is something both its modern historians agree on, and they come at the Nullification Crisis from very different angles.

7. Every previous great sectional crisis had this same cast of characters and all of those are all about slavery. Did some unseen, unmarked total revolution in American sectional politics happen between 1854 or 1858 and 1860 that de-centered slavery? Given the final antebellum decade is one of rapidly increasing tension specifically over slavery, that would be one hell of a trick.

8. Most damningly and dishonestly, "southern agriculture" differs from Northern agriculture exclusively in that the cash crops are actually grown by the nation's four million slaves. The money that Southern whites think they're losing to the tariff is actually money they're ripping out of the bodies of the black Americans they enslave, rape, and torture.

Now let's look at the sources. As you note, Jefferson Davis emphasizes economic reasons. The economic system of the South is slavery, which Davis and everyone who read him knew full well. Here he goes again:

The declared purpose of the compact of Union from which we have withdrawn was "to establish justice, insure domestic tranquillity, provide for the common defense, promote the general welfare, and secure the blessing of liberty to ourselves and our posterity;" and when, in the judgment of the sovereign States now composing this Confederacy, it had been perverted from the purposes for which it was ordained, and had ceased to answer the ends for which it was established, a peaceful appeal to the ballot-box declared that so far as they were concerned, the government created by that compact should cease to exist

What threatened Southern domestic tranquility? A Republican victory. Why would that do so when a Whig victory had not? The Whigs, while they had an antislavery wing, were not an antislavery party. The GOP was from its inception and as its defining trait, antislavery. Davis doesn't mean just chilling out here when he refers to tranquility. He's riffing on the belief that antislavery politics lead to slave revolts by inciting slaves, who totally love everything white enslavers do for them, to insurrection. They actually argued this.

Speaking of, let's look at the full paragraph from your quote:

An agricultural people, whose chief interest is the export of a commodity required in every manufacturing country, our true policy is peace, and the freest trade which our necessities will permit. It is alike our interest, and that of all those to whom we would sell and from whom we would buy, that there should be the fewest practicable restrictions upon the interchange of commodities. There can be but little rivalry between ours and any manufacturing or navigating community, such as the Northeastern States of the American Union. It must follow, therefore, that a mutual interest would invite good will and kind offices. If, however, passion or the lust of dominion should cloud the judgment or inflame the ambition of those States, we must prepare to meet the emergency and to maintain, by the final arbitrament of the sword, the position which we have assumed among the nations of the earth. We have entered upon the career of independence, and it must be inflexibly pursued. Through many years of controversy with our late associates, the Northern States, we have vainly endeavored to secure tranquillity, and to obtain respect for the rights to which we were entitled. As a necessity, not a choice, we have resorted to the remedy of separation; and henceforth our energies must be directed to the conduct of our own affairs, and the perpetuity of the Confederacy which we have formed. If a just perception of mutual interest shall permit us peaceably to pursue our separate political career, my most earnest desire will have been fulfilled. But, if this be denied to us, and the integrity of our territory and jurisdiction be assailed, it will but remain for us, with firm resolve, to appeal to arms and invoke the blessings of Providence on a just cause.

The bolded portion is Davis spelling out that differences in economic philosophy are basically irrelevant. He fears Yankee domination. But why would he fear specifically domination by free states instead of, say, populous states like Virginia? (Virginia was not in the CSA at the time of Davis's speech.) If there's no essential rivalry between the economic systems -and there's not, the northern and southern economies were pretty integrated by period standards- then he's obviously not worked up about factory men taking over because they're factory men.

But I press on:

With a Constitution differing only from that of our fathers in so far as it is explanatory of their well-known intent, freed from the sectional conflicts which have interfered with the pursuit of the general welfare, it is not unreasonable to expect that States from which we have recently parted may seek to unite their fortunes with ours under the government which we have instituted.

The states Davis expects are the Upper South slave stats that haven't chosen insurrection at the time of his speaking.

Sectional conflicts are slavery conflicts and the reference to the Confederate constitution is a tell you could see from space. It is, in most respects, essentially a language update of the familiar US constitution. Where it differs, it does so to make some rhetorical points, on a few details, and in making slavery permanently protected and obligatory upon the entire Confederacy, no matter what, forever. That's the original intend Davis is lauding as now updated.

Davis' speech is thick with doubletalk like this, which is why it's such a popular neo-Confederate text. If you don't know the era, the cant completely obscures what the Cornerstone Speech and the Declarations of Causes make clear.

This is also Jefferson Davis, making a major speech on the causes of the war to the Confederate congress, April 29, 1861, as quoted in Dow's Apostles of Disunion

The northern Majority in Congress had engaged in "a persistent and organized system of hostile measures against the rights of the owners of slaves in the Southern States [...] Fanatical organizations ... were assiduously engaged inciting amongst the slaves a spirit of discontent and revolt; means were furnished for their escape from their owners; and agents secretly employed to entice them to abscond," he insisted. The Republican Party, gripped by "a spirit of ultra fanaticism," was determined to deny slave owners access to the territories and would surround the South with "states in which slavery should be prohibited...thus rendering property in slaves so insecure as to be comparatively worthless." Davis described slavery itself as an institution which "a superior race" had used to transform "brutal savages into docile, intelligent, and civilized agricultural laborers," now numbering close to 4 million chattel in the South. "With interests of such overwhelming magnitude imperiled, the people of the Southern States were driven by the conduct of the North to the adoption of some course of action to avert the danger with which they were openly menaced"

Jefferson Davis, meet Jefferson Davis. Who should you believe? Both of them, but not the Davis after the war that tries to burnish the slaveholder's rebellion with high principle all over again.