After a bit more searching, here's something closer to an origin of the song... not clear how much of the story is 'genuine' folk or just someone's tall tale of years past, but here's something: https://www.cantonrep.com/article/20071222/NEWS/312229963
The similarities to the lines provided to the New Christy Minstrel version gives me doubts, but perhaps it's a starting place.

Jim Hillibish: No one can forget Jimmy’s Mule

Cairo in northern Stark County refuses to blow away. Sometimes it’s folk lessons that are the glue.

In the mid-1800s, everybody knew Cairo. Rich loads of coal in the vicinity attracted haulers, first in mule wagons, then steam trains to Canton. Buck Morgan insisted there was so much coal, it jumped into your buckboard as you passed.

In the memories of some is a coal-mining creature named Jimmy’s Mule.

One day, as Jimmy and Jimmy’s Mule were awaiting something, Jimmy bought a jar of 20-cent white lightning from a miner. He handed him a quarter and the purveyor dutifully and purposefully counted him three filthy cents in change. It was almost grand larceny, and Jimmy didn’t know it.

Jimmy’s Mule had taken all of this in with the wariness of a March hay merchant. He raised such a heavy snort, our moonshining friend forked over the other two coppers and fled with a mule bite ripping drawers and flesh of his rear end.

That night, the boys, over the ’shine, birthed the Legend of Jimmy’s Mule, which exists to this day:

Jimmy’s Mule can tote a wagon, he can pull a plow.
He can figure ’rithmetic and milk a muley cow.
No need no education. Ain’t never sat no schooool.
Anything worth doin’ can be done by Jimmy’s Mule.

It seemed to please that old coot of a mule tied to a mine ventilator to hear that git-fiddle and those words drift over the flat Cairo plains. If mules could smile, and some say they do on the insides, Jimmy’s Mule did.

A few days later, some federal mine inspectors cast a cloud over town. They wanted records, tax receipts, everything owner Buck Morgan had figured he’d get to filling out tomorrow but never did. The revenue boys had him by the coal tongs, that is, until they crossed the yard shoving Buck ahead of them toward his shack of an office. Jimmy’s Mule, up for watering and haying, shot across the yard. While connecting rear hooves to federal man flesh on one of those boys, he flailed incisors at the other, eating a hearty portion of the man’s shirt and heading in toward meat and bone.

Funny how those federals suddenly figured old Buck had everything in order. They were last seen running past Medina, tatters flapping.
That night over some shine, this time provided by Buck, the Legend expanded a piece:

I got fed up the other day at paying my federal rent.
I called up the White House; I got the pres-i-dent.
If you don’t cut out this mining tax, you ain’t no friend of mine.
He said, I’ll con-sult with Jimmy’s Mule, he’s on the other line.

This song does go on and on, built over decades of celebrating the mule that could do anything.

Let it be said that in addition to being the first sterile mule to father 157, Jimmy’s Mule took credit for inventing the steam-mining machine, sitting in the State Legislature for six terms and bagging limit deer within 10 minutes of the start of every hunting season.

All this remained comfortably planted in two-part harmony with plentiful fiddle and harmonica chances before each chorus, of course.

As was the usual case, Buck’s mine soon got too deep to work coal. Buck sold everything off, including Jimmy’s Mule. It didn’t mean much anyway, since all the miners were joining the Army. There was war in Europe. It was 1918.

As for what happened to Jimmy’s Mule, nobody seems to know. Some say there’s an epitaph on a grave near a battlefield in perhaps France that sheds some light.

It memorializes an unnamed ammunition mule, a critter that braved shell, bullet and gas without fail, lugging the fight to our boys in the trenches, returning with litters of their torn bodies:

Here lies Jimmy’s Mule of Cairo, Ohio

He kicked 27 privates, 19 corporals, 11 sergeants, 10 lieutenants, 7 captains, 5 majors, 3 colonels, and one box of howitzer shells.

He didn’t need no education, he never went to school.

Anything worth doing can be done by Jimmy’s Mule.

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(Note that though Jimmy Martin does a song called Jimmy's Mule, it's not this song but rather a version of Johnson's Old Gray Mule)