The nesting count would not be the hard part with the suggested syntax but arranging for \mydelimiter to swap between open and close delimiter depending on being prefixed by \left or \right would probably involve redefining the \left and \right primitives and have bad consequences somewhere..
So here is the same idea with a modified syntax that is simpler to use and a lot simpler to implement.
As noted in comments using \left \right in these cases is not always optimal and introduces additional horizontal space even when the delimiters are not enlarged.

\documentclass{article}
\def\ldelim{[}
\def\rdelim{]}
\def\zlefts{\left[\def\zright{\right]}\let\zleft\zleftr}
\def\zleftr{\left(\def\zright{\right)}\let\zleft\zlefts}
\let\zleft\zlefts
\begin{document}
$x = \zleft \zleft a+b \zright * y \zright / z $
$x = [( a+b ) * y ] / z $
\end{document}
$x = \\myleft \myleft a+b \myright *y \myright / z $(leftand\rightare delicate beasts) – David Carlisle Jun 04 '19 at 15:59\leftand\righthttps://tex.stackexchange.com/questions/173717/is-it-ever-bad-to-use-left-and-right – David Carlisle Jun 04 '19 at 16:00