I recommend the REALLY old country, the genre spawned by the Carters and Jimmie Rodgers in the early days of radio. That really was folk music, and the pioneers achieved stardom solely because the same music they had been making with friends and family in weekend parlors (or riding the rails desperately seeking work) suddenly had a means of broadcasting it nationwide. The only difference between what they had been doing recreationally all their lives and what they the began doing professionally was that they started doing it in the parlor of millions of new radio owners rather than just their own.

That early country music has relatively few glorifications of honky-tonking (which would have alienated its largely Southern Baptist audience—and artists!) pickups or whoring around town. Most of it is pastoral stuff like the song you linked, ballads in the Scottish tradition brought over two centuries earlier (often strikingly reminiscent of the kind of music you linked here,) gospel or hobo music like Jimmie Rodgers produced.

The "western" element embodied in "country and western" does not appear much until the '40s as we start hearing more about TX and OK, and less about TN and AL, but there is plenty of good stuff to be found there, too. Of the lot, I prefer Ernest Tubb most. Junior Brown reminds me a lot of him, partly because of his look, partly because of his use of steel guitar and partly because of the influence of Hank Thompson, who himself was largely influenced by Ernest Tubb. The Texas Troubadour himself was largely inspired by Jimmie Rodgers. Hank Thompson is not bad in his own right, despite being among the honky tonk pioneers. Hank Snow gives ET a run for his money with me, but at the end of the day TX>NS, especially when it comes to country music. Unless one considers Bluegrass a separate genre (many once did not) Bill Monroe is a must for country music fans as well, as are his former band duo Lester Flatt and Earl Scruggs.

It all starts with Jimmie Rodgers and the Carter Family though, at least among recorded music. Thence to Hank Snow (who probably did more than his co-production partner Tom Parker to bring the world Elvis) and Ernest Tubb, both of whom were heavily influenced by and credit Jimmie Rodgers. Chet Atkins was a musical virtuoso who seemed to think of himself more as a jazz guitarist (there is a great degree of overlap there, such as with Les Paul, or Billy Bird, whom ET cues by name on many recordings.) Atkins cites Merle Travis, another outstanding country artist of the '30s and '40s, highly among his own influences. Hank Williams is obligatory, but real country music was near the end of its run by the time he and Hank Thompson arrived; they were more than honky tonkers, but produced the finest examples of it and are primarily remembered for that.

Incidentally, from a purely musical standpoint, I highly recommend picking up some of the collaboration albums done by Chet Atkins (AKA Mr. Guitar.) There were two with Hank Snow alone, two more with Les Paul (frightening as that is,) numerous others with Jerry Reed and Homer and Jethro, and individual albums with various artists (my personal favorite being Neck and Neck with Mark Knopfler.)