I can speak for Anglo-American "folk music" (which is a very vague and not particularly useful category), and I would say there is little correlation between the song structure you provide and our oldest surviving examples. The "big ballads" are stories with a melody, and they can last from anywhere between a handful of verses to over a hundred. While they are very flexible pieces (the words, tune, and structure will readily shift based on the singer's preference, artistry, or memory), they employ short refrains in the middle or the end of a verse instead of a full chorus, which would only serve to inhibit the flow of the narrative.

Consequently, the ballads are often structured quite differently compared to modern songs. Riddles Wisely Expounded might be sung in rhyming couplets broken up by a semi-sensical refrain:

There was a lady of the North Country,
Lay the bent to the bonny broom,
And she had lovely daughters three.
Fa la la la, fa la la la ra re.

Some of the Robin Hood ballads, however, might place the refrain in the second line of the five-line verse:

In Nottingham there lives a jolly tanner,
With a hey down down a down down
His name is Arthur a Bland;
There is nere a squire in Nottinghamshire
Dare bid bold Arthur stand.

The structure of the verse can change even within a single performance, often as a deliberate artistic choice to emphasize a point in the story. One extreme example is this rendition of Jocky o Brineysland (known academically as Johnnie Cock) from the traditional singer Maggie Stewart. She allots most verses five lines, with the fourth line repeated as a refrain, except for several verses that she extends to six, seven, and nine lines.

Of course, the Anglo-American ballads are only one portion of "European folk music," or even Anglo-American "folk music." The three examples I've given here have their origins hundreds of years in the past; songs that had their genesis in more recent centuries are often less grand in scope and somewhat more likely to support a full chorus. I can't, however, think of any examples that follow the specific structure you provide.

Nearly all music has sections that repeat in some way. Repetition is the heart of structure, and without structure it no longer stands as a cohesive piece of music.

The verse-chorus structure became really popular with the 60s music, but similarities in the structure go back to operatic arias.

The Call and Response structure is cited as distinctively African, but once the response gets to be a bit longer, it's practically a chorus.

It's kind of hard to say that it is exclusively one or the other.

As with most things American, I think it's a bit of both. This structure has its roots in the 32 bar form of the "American songbook" i.e. the work of the tin-pan alley composers. This form has the basic structure of aaba (where 'a''s are like musical material and 'b' is different). This structure formed what was then called the 'chorus' and was often introduced by a 'verse' that had a somewhat freer form. The relationship between the verse and chorus was somewhat analogous to the relationship between a recitative and aria in classical opera. Gradually, the verse got slowly cut out or integrated in the structure of the song itself.

On the flip-side the modern song is also quite indebted to the blues, which is derived from the call and response structure of much African folk music. The Blues form uses an aab structure over twelve bars (rather than 32). The Blues are a much more narrative form, so stanza differ to a much greater degree than in 32-bar form. However, over time, the structure and melodies became more rigid and the b at the end of the form started becoming something of a tag-line or proto-chorus.

So the modern song has influences from both those genres. One can see the influence of the 32-bar form in the aabab structure, as well as influence from the blues in both the individual stanza structure and the idea of putting your tag at the end. However, the modern song form expands a great deal on these earlier forms and has it's own history of development that shaped it into what it is today.

One important note: I chose to start with the 32-bar form and the 12-bar Blues because they are the direct ancestors of this form (even though they're still alive and kicking). Both these forms are in fact American forms and even though I labeled them broadly as 32-bar being more European and Blues more African, they are both already mixed genres in which African and European influences collide.

Source: Most is picked out of my brain, but either sourced from J. Peter Burkholder's "History of Western Music" or my classes in pop-music history and pop-music analysis.