Put code snippets in a rounded gray frame?

Question

I would like to highlight code snippets in a grayish background with a rounded frame. I can achieve this partially with mdframed, but the caption is inside the frame.

I also tried with codebox with:

\newtcblisting{codebox}[1]{
    arc=4pt,
    colback=GitHubGray,
    boxrule=0.5pt,
    breakable,
    listing only,
    listing options={
        caption={#1},
        label=lst:label
    }
}

However, the solution isn't very pretty. I would rather prefer a traditional caption that I can use with \listoflistings later.

\documentclass{article}
\usepackage{lipsum}
\usepackage{color}
\usepackage{listings}
\usepackage[framemethod=tikz]{mdframed}
\definecolor{GitHubGray}{rgb}{0.9,0.9,0.9}
\lstset{
    language=c,
    breaklines=true,
    keywordstyle=\bfseries\color{black},
    basicstyle=\ttfamily\color{black}\fontsize{9pt}{10pt}\selectfont,
    emphstyle={\em \color{gray}},
    keepspaces=true,
    showspaces=false,
    showtabs=true,
    tabsize=2,
    upquote=true,
    aboveskip=2pt,
    belowskip=2pt,
    framexleftmargin=2pt,
    extendedchars=true,
    inputencoding=utf8,
}
\mdfsetup{%
  roundcorner=4pt,
  innerleftmargin=2mm,
  innertopmargin=4.5mm, % Magical value
  %linecolor=black!50,%
  innerbottommargin=1mm, % Magical value
  backgroundcolor=GitHubGray
}
\begin{document}
\section{Example}
\lipsum[1]
\begin{mdframed}
\begin{lstlisting}[caption={caption text},label=lst:label]
#include <stdio.h>
int main(int argc, char **argv) {
    printf("Hello, world!\n");
}
\end{lstlisting}
\end{mdframed}
\lipsum[2]
\end{document}

Is there a way to achieve this without the magical constants (top/bottom margin) and ideally less boilerplate (no additional \begin/\end something before and after the lstlisting.

Is that possible?