Yes, I would agree that the regular and even modulation of pitch we call “vibrato” usually emerges through vocal training (although, vibrato itself is not specifically trained). It is definitely true that untrained singers rarely exhibit the regularity, constancy and evenness of vibrato in trained singers. I also agree that classical singing, like any physical activity, depends in part on developing the natural attributes of the human body beyond where they would be without training.
That said, one of the points I have been making is that some modulation of pitch is a natural attribute of the human body. This is because modulation in muscle activity (as opposed to sustained tension) is natural to the human body. This is not to say that people don’t have chronic muscle tension, but it is to say that the body isn’t designed to work that way. Singers allow this specific attribute of the vocal musculature to become regular and even through training by eliminating various inhibiting tensions and regulating the source material. But, as I said before, it is not difficult to get a singer to produce an uneven, irregular and intermittent vibrato. IMO, one of the reasons so few untrained singers in America employ vibrato singing is that the modern pop aesthetic does not value this sound and therefore vibrato is routinely inhibited. In cultures where vibrato is a valued part of singing, most singers do seem to use vibrato. It is also not insignificant that the vast majority of singers who sing without electronic amplification (and, usually, electronic modification/improvement of tone) use vibrato.
The fact that vibrato is an innate attribute of the voice instrument makes it fundamentally different from vibrato on any other instrument. Variation in pitch is not an innate attribute of the violin, the trumpet or the clarinet. A cellist can develop his/her technique to the fullest, but the vibrato will never be there unless the cellist physically and consciously makes it happen. This is what I mean by “natural.”
As far as the sports analogy goes… it is true that a four minute mile is not exactly “natural.” However, I’m not sure this is a good comparison to vibrato. A better comparison to sprinting might be a tenor singing a full-voice high C. This is something that is not exactly unnatural, but it’s not exactly natural either. It depends on developing a natural ability of the body beyond what it would normally be. While the potential to produce a full-voice high C may be innate to certain voice instruments, the high C itself is not and must be developed. Vibrato, on the other hand, doesn’t work quite the same way. IMO, a better sports analogy to vibrato would be athletes who develop their running technique so that the body is working most efficiently for running. The body has a certain way that is best for running that depends on certain innate attributes of the human organism. The correct running motion isn’t something that has to be developed, it’s something that’s naturally there in the way the human body is put together.
Perhaps the best analogy that illustrates the difference would be to body building and Alexander technique. Body building is not exactly unnatural, but the human body really isn’t designed to carry that much muscle. It’s developing a natural attribute of the body beyond it’s natural state. In this way, body building is like sprinting and high note singing. Alexander technique, on the other hand, is learning to eliminate certain habits and develop others that together allow the body to function more efficiently according to how it is designed. This is allowing a natural attribute to work uninhibited in its natural state. In this way, Alexander technique is like running form and vibrato. One does not have to “build up” a vibrato or the correct way to lift a box the way one has to build up a high C or a sprinter’s legs.