Problem converting equations to images on Windows 10

Question

My problem follows on from the question posed by JansthcirlU and from the response by Marijn (Converting equations to images). I blindly copied Marijn's first MWE and it works perfectly on Overleaf. But I can't use it on Overleaf for my own project because of the compilation timeout, so I tried on my local installation (TexLive 2023 + TexMaker) and I'm getting the following error :

LaTeX Warning: File '{main-figure0}.png' not found on input line 53. ! Package pdftex.def Error: File 'main-figure0.png' not found: using draft setting.

Equations are well converted to pdf, but the pdf are not converted into png and I have 4 differents files for each equation : .dpth, .md5, .log and .pdf. As Marijn said, since I'm on Windows, I replaced rm -f to del and convert to magick inside the MWE but it still doesn't work. I also have the latest ImageMagick release installed and when I type where magick in Windows' cmd, I got the right path.

What do you think is the problem ?

Edit 1 : Here is the content of my .log file :

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LaTeX Warning: File `{main-figure1}.png' not found on input line 60.
! Package pdftex.def Error: File `main-figure1.png' not found: using draft sett
ing.
See the pdftex.def package documentation for explanation.
Type  H <return>  for immediate help.
 ...
l.60 }
Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.
Overfull \hbox (5.2577pt too wide) in paragraph at lines 60--61
[][]
 []
Underfull \hbox (badness 10000) in paragraph at lines 60--61
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[1
{c:/texlive/2023/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] (./main.aux)

LaTeX2e <2023-11-01> patch level 1
L3 programming layer <2024-02-20>

) 
Here is how much of TeX's memory you used:
 13144 strings out of 474136
 272862 string characters out of 5748496
 1927190 words of memory out of 5000000
 35268 multiletter control sequences out of 15000+600000
 559296 words of font info for 41 fonts, out of 8000000 for 9000
 1141 hyphenation exceptions out of 8191
 102i,5n,107p,541b,241s stack positions out of 10000i,1000n,20000p,200000b,200000s
<c:/texlive/2023/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb><c:/tex
live/2023/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb><c:/texlive/2023
/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb><c:/texlive/2023/texmf-dis
t/fonts/type1/public/amsfonts/cm/cmr10.pfb><c:/texlive/2023/texmf-dist/fonts/ty
pe1/public/amsfonts/cm/cmtt10.pfb>
Output written on main.pdf (1 page, 57436 bytes).
PDF statistics:
 36 PDF objects out of 1000 (max. 8388607)
 22 compressed objects within 1 object stream
 0 named destinations out of 1000 (max. 500000)
 13 words of extra memory for PDF output out of 10000 (max. 10000000)

Edit 2 : I've partially solved the problem. It seems that Ghostscript is needed in order to rasterize vector files such as PDF (https://stackoverflow.com/questions/32466112/imagemagick-convert-pdf-to-jpeg-failedtoexecutecommand-gswin32c-exe-pdfdel). Now, when I try to manually convert a .pdf equation (main-figure0.pdf for example) using magick -density 300 "main-figure0.pdf" "main-figure0.png" in cmd, it's working. But the MWE still doesn't work, so it seems that magick command can't be executed inside a LaTex script and I don't know why. However, when I manually converted the .pdf files to .png files, the script ran smoothly. Here is the actual MWE with modifications to work on Windows :

\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{external}
\tikzexternalize
\tikzset{%
    % Add size information to the .dpth file (png is in density not size)
    /pgf/images/external info,
    % Use the png export AND the import
    use png/.style={png export,png import},
    png export/.style={
        external/system call=%
        {pdflatex \tikzexternalcheckshellescape -halt-on-error -interaction=batchmode -jobname "\image" "\texsource";%
         magick -density 300 "\image.pdf" "\image.png"; del "\image.pdf"},
    },
    png import/.code={%
        \tikzset{%
            /pgf/images/include external/.code={%
                % Here you can alter to whatever you want
                % \pgfexternalwidth is only available if /pgf/images/external info
                % is set
                \includegraphics%
                [width=\pgfexternalwidth,height=\pgfexternalheight]%
                {{##1}.png}%
            }%
        }%
    }%
}
\newcounter{pngeq}
\setcounter{pngeq}{0}
\newcommand{\pngequation}[1]{%
\tikzset{use png}
\begin{tikzpicture}
\node (eq) {
\hspace{-1.1\parindent}\begin{minipage}{\textwidth}%
\setcounter{equation}{\number\value{pngeq}}
\begin{equation}
#1
\end{equation}
\end{minipage}
};
\end{tikzpicture}%
\stepcounter{pngeq}
}
\begin{document}
\section{Linear equation through a point given its gradient}
A very simple differential equation is to find a linear equation through a particular point $P(x_i,y_i)$ given its gradient $k_i$ (or more commonly used for lines, its slope).
\pngequation{
\frac{dy}{dx} = k_i \iff \int dy = \int k_i \cdot dx \iff y = k_i \cdot x + C
}
Plug in the initial conditions to find the particular value of the constant of integration $C$.
\pngequation{
\frac{dy}{dx} = k_i \iff \int dy = \int k_i \cdot dx \iff y = k_i \cdot x + C'
}
\end{document}

If anyone can spot why the script does't want to convert .pdf file to .png file, I'm all ears.