I'll attempt a scientific answer which might partially explain this, but I should also say up front that I believe there is a large social/cultural component to this which is explained well in the other answers.
There have been a number of studies (see below) which asked people to rank particular dyads (two notes together) and triads (three notes - commonly known as a chord) in terms of consonance and dissonance. (See Kameoka and Kuriyagawa, 1969; Hutchinson and Knopoff, 1978, 1979; Cook and Fujisawa, 2006; Johnson-Laird et al., 2012)
A couple of things arise from those studies. Major chords are almost always at the top of a ranked preference list for consonance, with minor chords coming in slightly lower. Another point is that the results differ between musicians and non-musicians with the latter being more tolerant of some kinds of dissonance or chord alterations.
Alongside these studies, the study of what makes a chord consonant or disonant has been studied since the ancient greeks. Even the great mathematician Leonard Euler wrote a lot on this in 1739. Of all of these theories that I've looked at, I find the recent work by Frieder Stolzenberg to be the most convincing, in particular the paper "Harmony Perception by Periodicity Detection". That paper contains details of the studies on chord preference and also has good references to the other work in this area.
The details of Stolzenberg's work aren't too important here but the point I want to emphasis is that we, as humans, seem to respond to some chords differently and that we naturally gravitate to those which are more periodic. This is backed up by a plausible neurological model on how our perception of harmony works. That suggests that harmony isn't a 100% cultural phenomenon.
How does that relate to country? Here's where I have to hypothesise a little bit because I'm not aware of any studies that address this directly. My impression is that the transmission and popularity of a country music song relied mainly on being passed from player to player initially and then, later, on its ability to make an impression on the first listen when broadcast by radio. The audiences would probably have mainly been without musical training. Given our general preference for consonant, major chords exists this suggests that songs which used a simpler set of chords would be more transmissible.
If you look at something like jazz, it seems plausible that the harmonic complexity of that music developed over time. This might be related to the way it was originally intended for dancing. Dancers wouldn't judge a song by the harmony alone, which gave the musicians (with their more developed palette of consonance and dissonance) more room to stretch out and embelish the harmony.
As an example, if you count Western Swing as country (which I do) you can hear that the harmony is slightly more complex than typical "cowboy songs" of the era (lots of 6ths and 9ths, some blues scales). This might be because Bob Wills and similar acts were known for dancing primarily before then gained wide appeal via radio.
As I said, this can only ever be a partial answer due to the complexity of what makes up a given music genre. I hope that this offers a direction on why a genre might prefer simple chords though.