The TRPV1 receptor is present on sensory nerves, so although it does primarily detect high temperature[1], regulating it can result in pain relief in general, not just a reduction in "burning sensations." Also, the article simplifies the rather complicated molecular biology, much of which is still being worked out. Importantly the article does not discuss killing nerves, just their nerve endings.

[1]TRPV1 also detects high acidity and the endocannabinoid anandamide.

https://en.wikipedia.org/wiki/TRPV1

PS: Your links [1] and [2] are identical... mistake?