But since Arizona (not great land for plantation agriculture) was the only territory the Confederates apparently even tried to claim, why was this position even worth seceding and fighting for?

Because that's not all the Confederates tried to claim. They wanted everything below 36°30' latitude to become a slave state. That would include New Mexico, Arizona, and southern California. And Congress could have split these areas into as many states as they wanted, so there was no guarantee that there would only be three more slave states entered into the Union.

But that wasn't really the sticking point. There were compromisors who were willing to go along with that, if that's all that was needed to resolve the issue. But the real issue is that the Confederates wanted more - at least enough to keep parity with the Northern territories that would be earmarked as free states.

The first peace effort during the Secession Crisis resulted in what is known as the Crittenden Compromise, which originated in the U.S. Senate. Here is the text of the proposed Constitutional amendment on the matter:

In all the territory of the United States now held, or hereafter acquired, situated north of latitude 36°30', slavery or involuntary servitude, except as a punishment for crime, is prohibited while such territory shall remain under territorial government. In all the territory south of said line of latitude, slavery of the African race is hereby recognized as existing, and shall not be interfered with by Congress, but shall be protected as property by all the departments of the territorial government during its continuance...

It's that "hereafter acquired" clause that was the sticking point. The Confederates didn't like the wording of the amendment, because it was vague on what happens below the slavery line. In February 1861, during the Washington Peace Conference, the state of Virginia proposed this alternative wording (emphasis mine):

In all the territory of the United States now held or hereafter acquired, situate north of latitude 36°30', slavery or involuntary servitude, except as a punishment for crime, is prohibited, while such territory shall remain under territorial government. In all the territory now or hereafter acquired south of said line of latitude, slavery of the African race is hereby recognized as existing, and shall not be interfered with by Congress; but shall be protected as property by all the departments of the territorial government during its continuance...

Does that make it more clear? The sticking point was that the Confederates were trying to get a clause into the Constitution that would guarantee that any newly-acquired land below 36°30' would become a slave state. This had very serious ramifications, in light of the Mexican-American War, the Ostend Manifesto, and other similar efforts. There was open discussion about purchasing or else invading Cuba, and other countries/colonies south of the border. With this clause, U.S. slavery could potentially extend all the way to the southern tip of South America, if the U.S. could acquire the land.

In his 1961 article on the Washington Peace Conference, historian Samuel Eliot Morison describes the debate over the clause this way, after the state of Virginia had proposed the above text:

The Virginia General Assembly, in issuing the call to the Convention, had resolved that the Crittenden Compromise, extending the 36°30' line to California, would be acceptable only if it protected slavery in all territory "now held or hereafter acquired" south of this line. Those three words "or hereafter acquired" were really Virginia's ultimatum. They meant, as everyone knew, that slavery could be extended into any future territory, such as Cuba, the northern tier of Mexican states, or Nicaragua, that might be acquired by purchase, filibustering, or war. That demand, as we shall see, was absolutely and completely unacceptable to the northern states; but the Virginia delegation would accept nothing less. [Virginia delegate] James Seddon said so, frankly, to [Massachusetts delegate George S.] Boutwell, when that head of the Massachusetts delegation called on him—"We must have new lands." There must have been many unrecorded conversations of this kind among the delegates, and between them and other politicians in Washington.

Morison goes on to say that Virginia's position was "rigid". When the above proposal was rejected, the Virginia delegation came back the following week with a "new" plan which said exactly the same thing on territorial expansion as did the original plan.

The same roadblock was encountered in the U.S. House during the peace efforts of the "Committee of Thirty-Three" that led to the Corwin Amendment. For instance, Rep. Henry Winter Davis of Maryland (who would later go on to become a Radical Republican and was instrumental in getting slavery abolished in his state) offered the following counter-proposal:

1. Arizona and New Mexico would be admitted into the Union as one big slave state. In exchange, California would remain intact as a free state.

2. Any southern lands "hereafter acquired" could retain slavery if slavery already existed there (e.g., Cuba) while the land remained a federal territory. But upon statehood and after, the state reserved the right to abolish slavery at will.

3. In exchange, no foreign lands could be purchased or invaded, or admitted into the Union as a state, without a 2/3 majority vote in both houses of Congress.

This was unacceptable to the Confederates, because that would mean they'd need Northern votes to get any of those "hereafter acquired" territories admitted into the Union as slave states. That was a tall order. Winter's proposal was not adopted.

In his introduction to the Committee's report, Rep. Thomas Corwin (R-Ohio) characterized the debate on the matter this way:

From the beginning of our deliberations, it was apparent that the disposition of that portion of our territory lying south of the parallel of 36°30' was the main subject of difficulty. The settlement of that question was, however, complicated with a provision much insisted on for territory hereafter to be acquired. This did not seem to the committee properly to belong to the subject. The committee did not think proper to extend their consideration of the embarrassments arising out of the occupation of territory now within our possession, to territory which might or might not hereafter be acquired.

It seemed to them improper, if not absurd, while our government was threatened with overthrow by an angry controversy touching the disposition of our present territorial possessions, to employ their time in arranging for a partition amongst ourselves of the territorial dominions of neighboring nations, looking to a future which, when it shall come, will probably bring with it circumstances and conditions which could not be now foreseen, and which, therefore, should be left to the judgments of those whose duty it may become to consider and act upon them.

On December 29, 1860, Corwin himself motioned to have the "hereafter acquired" clause struck out of the text of the Committee's compromise proposal; the motion passed 17 to 10. In reaction, Rep. Miles Taylor (D-Louisiana) boycotted any future votes (and would leave Congress soon after when Louisiana seceded):

After the vote on Mr. Corwin's amendment to strike out the words "or hereafter acquired" in article 1 of the joint resolution of Mr. Nelson had been determined in the affirmative, Mr. Taylor asked and obtained leave to have the following recorded on the journal:

Mr. Taylor remarked that the decision thus reached made it clear to his mind that there would be no agreement by the committee upon propositions for the adoption of amendments to the existing Constitution which would be effectual for the settlement of the issue now pending between the two great sections of the country growing out of the slavery question, and that, in consequence, it was his purpose to take no further part in the deliberations of the committee.

Why did this ability to expand slavery matter so much to the South? Simple. Because they wanted to expand slavery. They wanted to keep parity in the U.S. Senate, and in the federal government in general, with the free North. Limiting expansion to only New Mexico and Arizona would put the South in the minority permanently. From Albert Pike's pro-secession pamphlet "From State or Province? Bond or Free?" published in Arkansas in early 1861:

...they [anti-slavery Northerners] have become too strong for us, outnumber us in people and States, receive a hundred thousand emigrants per annum, have open space wherein to make six or seven new States, and announce it as their ultimatum that we shall not expand southwardly or southwestwardly, but slavery shall be prohibited in all territory hereafter acquired.

He who reflects on all this, cannot well fail to see that separation was but a question of time. Sooner or later the North and South could not help but divide...

Without getting the concessions on the issue from the North that they desired during any of the peace negotiations, the Southern states continued to secede.

SOURCES:

The Report of the Committee of Thirteen that led to the Crittenden Compromise.

The Report of the Committee of Thirty-Three that led to the Corwin Amendment.

The Report of the Washington Peace Conference

"The Peace Convention of February, 1861" by Samuel Eliot Morison, Proceedings of the Massachusetts Historical Society, 1961

Southern Pamphlets on Secession, November 1860-April 1861, ed. by Jon L. Wakelyn, 2000