According to my experiments, actual (La)TeX also gets \not\vert wrong: It does produce two crossing lines, but with quite strange spacing surrounding them.

Per this tex.SE question, \vert is not intended to produce a symbol with relation spacing at all, so arguably an author has no right to expect \not\vert to do anything meaningful in the first place.

The existence of \nmid probably relates to the fact that \not\mid gets it completely wrong even in actual (La)TeX.

(The linked tex.SE answer claims that there is a \divides, but neither MathJaX nor LaTeX-out-of-the-box recognizes that at all, and I'm not sure which package it is from -- neither amsmath nor amssymb do the trick).

According to my experiments, actual (La)TeX also gets \not\vert wrong: It does produce two crossing lines, but with quite strange spacing surrounding them.

Per this tex.SE question, \vert is not intended to produce a symbol with relation spacing at all, so arguably an author has no right to expect \not\vert to do anything meaningful in the first place.

The existence of \nmid probably relates to the fact that \not\mid gets it completely wrong even in actual (La)TeX.

(The linked tex.SE answer claims that there is a \divides, but neither MathJaX nor LaTeX-out-of-the-box recognizes that at all, and I'm not sure which package it is from -- neither amsmath nor amssymb do the trick).