Remove bullet and leading space before list item

Here is a solution:

\documentclass[10pt]{article}
\usepackage{enumitem}
\newlist{balditemize}{itemize}{1}
\setlist[balditemize]{label=, wide=\parindent, labelsep*=0pt, leftmargin=*, topsep=0pt, itemsep=0pt}
\usepackage[showframe]{geometry}

\begin{document}

\section{Section Title}
\subsection{Subsection Title}

\begin{itemize}[label={$\bullet$}, topsep=0pt, itemsep=0pt]
    %
    \item List item with bullet that I want gone
        \begin{itemize}[label={$\bullet$}, topsep=0pt, itemsep=0pt]
            \item Inner list item (I want to keep these bullets)
            \item Inner list item (I want to keep these bullets)
        \end{itemize}
    %
    \item List item with bullet that I want gone
        \begin{itemize}[label={$\bullet$}, topsep=0pt, itemsep=0pt]
            \item Inner list item (I want to keep these bullets)
            \item Inner list item (I want to keep these bullets)
        \end{itemize}
    %
\end{itemize}

\section{Section Title}
\subsection{Subsection Title}

\begin{balditemize}
   \setlist[itemize, 1]{wide=\dimexpr\leftmargini-\fontdimen2\font, leftmargin=*, labelsep =5pt, topsep=0pt, itemsep=0pt}%
    \item List item with bullet that I want gone List item with bullet that I want gone List item with bullet that I want gone List item with bullet that I want gone
        \begin{itemize}
            \item Inner list item (I want to keep these bullets) Inner list item (I want to keep these bullets) Inner list item (I want to keep these bullets) Inner list item (I want to keep these bullets)
            \item Inner list item (I want to keep these bullets)
        \end{itemize}

    \item List item with bullet that I want gone
        \begin{itemize}
            \item Inner list item (I want to keep these bullets)
            \item Inner list item (I want to keep these bullets)
        \end{itemize}
    %
\end{balditemize}

\end{document} 

You might want a more automatic approach, but here is an ad-hoc solution:

You could use \item[] combined with leftmargin. By inspecting the values, the default leftmargin seem to be 25pt, and the width of the \bullet seem to be 5pt and the spacing between the bullet and the text also 5pt. Thus the new leftmargin should be 25pt - 10pt=15pt. Also we have to add 5pt to the "inner" left margin (5pt+25pt=30pt) to make up for the distance removed between the bullet and the item: and it actually works as expected:

\documentclass[10pt]{article}
\usepackage{enumitem}
\begin{document}

\section{Section Title}
\subsection{Subsection Title}
\begin{itemize}[label={$\bullet$}, topsep=0pt, itemsep=0pt]
    %
    \item List item with bullet that I want gone (below)
        \begin{itemize}[label={$\bullet$}, topsep=0pt, itemsep=0pt]
            \item Inner list item (I want to keep these bullets)
            \item Inner list item (I want to keep these bullets)
        \end{itemize}
    %
    \item List item with bullet that I want gone (below)
        \begin{itemize}[label={$\bullet$}, topsep=0pt, itemsep=0pt]
            \item Inner list item (I want to keep these bullets)
            \item Inner list item (I want to keep these bullets)
        \end{itemize}
    %
\end{itemize}

\begin{itemize}[label={$\bullet$}, topsep=0pt, itemsep=0pt, leftmargin=15pt]
    %
    \item[] List item with bullet that I want gone
        \begin{itemize}[label={$\bullet$}, topsep=0pt, itemsep=0pt, leftmargin=30pt]
            \item Inner list item (I want to keep these bullets)
            \item Inner list item (I want to keep these bullets)
        \end{itemize}
    %
    \item[] List item with bullet that I want gone
        \begin{itemize}[label={$\bullet$}, topsep=0pt, itemsep=0pt, leftmargin=30pt]
            \item Inner list item (I want to keep these bullets)
            \item Inner list item (I want to keep these bullets)
        \end{itemize}
    %
\end{itemize}
\end{document}

Output