The mathtools package can automatically insert italics correction when using \(...\) for inline math mode. See this answer. However, I much prefer $...$ for inline math as I find it more readable and comfortable to type on my keyboard layout.
I am currently using the following hack to keep using the $...$ syntax while preserving the italics correction.
\catcode`\$=13%\def$#1${\(#1\)}%I haven't had issues with it in my current document, but I was wondering whether there are reasons this might be a bad idea. I would image that if this was harmless the mathtools package would use it to also patch the $...$ syntax for the mathic feature.
So my question is: Is this safe and are there any caveats I might be overseeing?