How to reset counter each section with secnumdepth set to 0

You can step the counter for section, but suppress its output:

\documentclass[
paper=a4,
DIV=15,
parskip=half
]{scrartcl}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% the problematic option %%%
\setcounter{secnumdepth}{1}
\makeatletter
\renewcommand{\@seccntformat}[1]{}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcounter{thatimportantc}[section]
\newcounter{randomcounter}

\newcommand{\callingcounters}{%
    \stepcounter{thatimportantc}\arabic{thatimportantc}~|~%
    \stepcounter{randomcounter}\arabic{randomcounter}
}

\begin{document}

\section{The first section}

In this section, you will observe that the counters are equal (as expected).

\callingcounters\\
\callingcounters

\section{The second section}

In this section, you will observe that the counters are not equal (as expected).

\callingcounters\\
\callingcounters

\section{Third section}

\callingcounters\\
\callingcounters

\section{4th section}

\callingcounters

\end{document}

enter image description here

This will need actions also on the headers, if you use them.

Alternative way: patch \section so it steps the relevant counter.

\documentclass[
paper=a4,
DIV=15,
parskip=half
]{scrartcl}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{etoolbox}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% the problematic option %%%
\setcounter{secnumdepth}{0}
\preto\section{\stepcounter{section}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcounter{thatimportantc}[section]
\newcounter{randomcounter}

\newcommand{\callingcounters}{%
    \stepcounter{thatimportantc}\arabic{thatimportantc}~|~%
    \stepcounter{randomcounter}\arabic{randomcounter}
}

\begin{document}

\section{The first section}

In this section, you will observe that the counters are equal (as expected).

\callingcounters\\
\callingcounters

\section{The second section}

In this section, you will observe that the counters are not equal (as expected).

\callingcounters\\
\callingcounters

\section{Third section}

\callingcounters\\
\callingcounters

\section{4th section}

\callingcounters

\end{document}

Today a new prerelease of KOMA-Script version 3.31 was published. It can be installed from the KOMA-Script website. With this version

\documentclass[
%paper=a4,% default
DIV=15,
parskip=half
]{scrartcl}[2020/06/02]
\usepackage[T1]{fontenc}
%\usepackage[utf8]{inputenc}% only needed with older TeX distributions
\usepackage{lmodern}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% the problematic option %%%
\setcounter{secnumdepth}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcounter{thatimportantc}[section]
\newcounter{randomcounter}

\newcommand{\callingcounters}{%
  \stepcounter{thatimportantc}\arabic{thatimportantc}~|~%
  \stepcounter{randomcounter}\arabic{randomcounter}
}

\begin{document}
\KOMAScriptVersion% show the KOMA-Script version used for the example
\tableofcontents
\section{The first section}
In this section, you will observe that the counters are equal (as expected).

\callingcounters\\
\callingcounters

\section{The second section}
In this section, you will observe that the counters are not equal (as expected).

\callingcounters\\
\callingcounters

\section{Third section}
\callingcounters\\
\callingcounters

\section{4th section}
\callingcounters

\end{document}

results in

enter image description here

If you only want to suppress the section number in document, page header and TOC, you can use:

\setcounter{secnumdepth}{\sectionnumdepth}
\renewcommand*{\sectionformat}{}% supress section number in document body
\renewcommand*{\sectionmarkformat}{}% supress section number in page header
\renewcommand*{\addsectiontocentry}[2]{\addtocentrydefault{section}{}{#2}}% supress section number in ToC

This works with really old versions too. Example:

\documentclass[
%paper=a4,% default
DIV=15,
parskip=half
]{scrartcl}
\usepackage[T1]{fontenc}
%\usepackage[utf8]{inputenc}% only needed with older TeX distributions
\usepackage{lmodern}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% the problematic option %%%
\setcounter{secnumdepth}{\sectionnumdepth}
\renewcommand*{\sectionformat}{}% supress section number in document body
\renewcommand*{\sectionmarkformat}{}% supress section number in page header
\renewcommand*{\addsectiontocentry}[2]{\addtocentrydefault{section}{}{#2}}% supress section number in ToC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcounter{thatimportantc}[section]
\newcounter{randomcounter}

\newcommand{\callingcounters}{%
  \stepcounter{thatimportantc}\arabic{thatimportantc}~|~%
  \stepcounter{randomcounter}\arabic{randomcounter}
}

\begin{document}
\KOMAScriptVersion% show the KOMA-Script version used for the example
\tableofcontents
\section{The first section}
In this section, you will observe that the counters are equal (as expected).

\callingcounters\\
\callingcounters

\section{The second section}
In this section, you will observe that the counters are not equal (as expected).

\callingcounters\\
\callingcounters

\section{Third section}
\callingcounters\\
\callingcounters

\section{4th section}
\callingcounters

\end{document}

Result:

enter image description here