Appendix Section Title without A in Front but Appendix Subsection Titles with A in Front

It would have helped if your MWE compiled without errors.

However, try this:

% appendixprob.tex  SE 579384

documentclass[12pt,notitlepage]{article}
%DeclareCaptionLabelFormat{AppendixTables}{Table A.#2} %% causes an error
usepackage{appendix} %% could be useful but not necessary

begin{document}

section{A section}

appendix
%begin{appendices}
renewcommand{thesection}{arabic{section}}
renewcommand{thesection}{hspace{0.5em}} % no number but align with section titles
renewcommand{thesubsection}{Alph{section}.arabic{subsection}}
section{Appendix}

%captionsetup{labelformat=AppendixTables} %% causes an error
setcounter{table}{0}

subsection{Subsection 1}
subsection{Subsection 2}
%end{appendices}

end{document}

It does what you want regarding the numbering of section{Appendix} but I have no idea if the positioning is what you want. It is up to you to deal with the numbering of floats.

Add manually the appendix.

documentclass[12pt,letterpaper]{article}

counterwithin{table}{section}

newcommand{singleappendix}[1]{%
  appendix
  section*{#1}
  addcontentsline{toc}{section}{#1}
  stepcounter{section}
}

begin{document}

tableofcontents

section{Introduction}

begin{table}[htp]
caption{This is a table caption}
end{table}

singleappendix{Appendix}

subsection{Subsection 1}
subsection{Subsection 2}

begin{table}[htp]
caption{This is a table caption}
end{table}

end{document}