Google Books searches find three strands of use of alrighty in the late nineteenth and early twentieth centuries: as a translation of despotes in biblical translation; as a form of pidgin English used by natives of other lands; and as an American English form that appears to have originated in
'Alrighty God'
The Calvinism Popularised (1879) instance of Alrighty looks to me to be an outlier. Even Henry Long, the author of that work, concedes what may have been the strongest objection to it, immediately after the language that ermanen quotes from it in the question above:
He.—But Alrighty has a comical and profane sound about it, displeasing to the religious taste.
I.—That is true because the word is new, and therefore unused, but if it had come into use half a millennium since, we should have hedged it round with the sanctions of Ex. xx. 7, in which case it would have been well adapted for use.
...I affirm that Alrighty is a better translation of Despotes than Lord is. Our word Lord is connected with loaf, and means loaf giver, like the Italian Frangipani. ... Words come spontaneously rather than by invention, or we might have too many, and the only misfortune to me is that Alrighty is invented, and, perhaps, ineligible for adoption, but I hold it is far more true a translation of despotes than Lord is when the question of literality is allowed due weight. At least it conveys my meaning.
A Dutch speaker quoted in Transactions of the International Dental Congress (1905) seems to echo Long's views:
Is it merely accidental that about the time when this law originated the forementioned philosophical thought came also to the surface again? Possibly not. And I am well informed that not only the German, but many men of today still preach to the "Almighty" and not to the "Alrighty."
But a Google Books search suggests that no one took Long's advice to replace Lord with Alrighty in their translations of the Bible into English.
'All righty' and 'alrighty' as dialect forms of 'all right'
Following Henry Long's book, the next-earliest instance of the term in a Google Books search appears as all righty and appears in Mary Owen, "The Taming of Tarias," a story set in 1840 in Kentucky, in The Century Magazine (December 1889), where the character using the wording is a young white woman named Jincy King:
"A-h!" said Tarias, with the long-drawn breath of hesitancy ; then quickly, as if a sudden purpose had sprung into her soul, "Yes, yes; truly will Louis and I be pleased to have him with us."
“All righty! Be sure an' come early, an' tell Louis—but nev' min'; I'll tell 'im myself when I see 'im.”
Similarly, Ellen Kirk, Dorothy and Her Friends (1899) uses the term as dialect speech by native English speakers:
Without doubt the fishermen were happy. They were emptying the water out of their boots; they were red in the face with the salt wind; they were clapping their hands to warm them, all the time joking with each other, and watching Captain Narraway, whom they has asked to be umpire, and who was sorting out the fish into nine piles; just so many of the biggest into each; just so many of the medium size; just so many of the smallest.
"All right?" asked Captain Narraway, when his task was finished.
All righty," shouted the fishermen, with a cheer.
The fishermen have names like Sam Perkins, Ben Coe, and Ned DeForrest, so they are not supposed to sound exotic.
The earliest instance of the single-word spelling alrighty in the sense of "all right" or "okay" is in Joel Williams, A Missourian in the Far West (1906), cited by ermanen above.
Harold Wentworth, American Dialect Dictionary (1944) has this interesting entry for all righty:
all righty. All right; — used chiefly by women. Cf. indeedy, Whatty?
1890s n.e.cent.Ind[iana] Grant Co[unty]. A friendly form. Rural. 1912 w.Ind[iana]. H can beat you all righty. 1916 Neb[raska]. 'Papa & I like him alrighty.' Fanny Hurst Metropolitan Mag. Jan. In widespread college use. 1930s nW[est] V[irgini]a. Chiefly by women. ...
Harold Wentworth & Stuart Flexner, Dictionary of American Slang, first edition (1960), has this usage note for the various spellings of all righty:
all rightie, all righty, all rightey, all rightee interj. All right. [Examples omitted.] Dial[ect] use since c1890. Fairly common c1935–c1940 ; now considered affected or cute.
Barbara Kipfer & Robert Chapman, Dictionary of American Slang, fourth edition (2007) has the following update:
all righty affirmation (Variations: rightie or rightee or rightey may replace righty) A humorous or deliberately cute and childish way of saying "all right"
'All righty' as pidgin English
One nineteenth-century instance of all righty is from Frederick Williams [Herbert Hamblen], On Many Seas: The Life and Exploits of a Yankee Sailor (1897):
Monday night, at about eleven o'clock, the man who was keeping lookout on the forecastle-head reported that our merchant was there, and two of us, after taking a look aft to see that the coast was clear, shouldered the two mattresses and carried them up, and looking over the bow we could dimly see the old fellow in his boat hanging on to the anchor chain ; so we hailed in a whisper, "All right, Johnny."
"All righty Chonny," was whispered back ; and making the end of a chest-lashing fast around them both, we lifted and shoved them over, and then, as they were heavy, we hung back and lowered away carefully until we felt them land.
Here the "all righty" is evidently an instance of pidgin English. So are the three instances in Owen Hall, "More Warm Than Pleasant," a story set in New Zealand, in The English Illustrated Magazine (January 1897):
"Plenty bath here! Plenty warm here! Taihoa, you all righty."
Mohi's manner didn't invite discussion, so I accepted the situation and made myself as comfortable as circumstances would permit at the driest end of the hut.
...
Mohi's last prescription was an undoubted success. When I tried to tell him so he only smiled gravely as he replied, "Kapai tihoa. You all righty by and by. Kapai Mokihinau!"
...
I had lain aside my staff, and when I returned to the hut without it I found Mohi sitting smoking at the entrance. "You all righty now?" he asked with a questioning uplifting of his heavy eyebrows.
The pidgin usage seems to have faded away as the twentieth century progressed and interest in representing the broken English of non-native English speakers became less prominent in published writing. In any case, it seems not to have drawn any dictionary coverage in recent decades.
Conclusions
The history of 'all righty' and 'alrighty' gets off on the wrong foot with the odd promotion of Alrighty as an epithet for God in Henry Long's religious polemic of 1879. But not long thereafter (in 1890), the term appears in its modern sense in a story set in Kentucky fifty years earlier.
Harold Wentworth reports that the term was in dialect from circa 1890 in Indiana (just north of Kentucky) and later elsewhere in the United States in the early 1900s, becoming college slang at some point. The Dialect Notes entry that ermanen cites in the opening question indicates that college use was widespread by 1913 (significantly earlier than Wentworth suggests), but still one or two decades after the term's initial rural use (probably).
Best Answer
You're right that "thanks" is always used as a plural noun, whereas when we say "thank you," the word "thank" is a verb.
You can explain that some nouns are always used in their plural form, and give other examples (clothes, outskirts, glasses, scissors, etc.).
You could alternatively tell your students each expression of thanks is a shortened form of a longer sentence that isn't really in use anymore.
(I) thank you. -- Thank you.
Thanks (be to you). -- Thanks.
or
(I give/feel) thanks. -- Thanks.