Alternatives to achieve certain output format

Enough simple?

\documentclass{report}
\usepackage{mhchem,parskip}
\def\a{\phantom{\ce{^12C + ^1H ->}}}
\def\b{\phantom{\ce{^13N ->}}}
\begin{document}
\ce{^12C + ^1H -> ^13N + \gamma}\par
\a\ce{^13N -> ^13C + $e^+$ + \nu}\par
\a\b\ce{^13C + ^1H -> ^14N + \gamma}
\end{document}

It's a matter of opinion whether this is less clunky...it eliminates all & tab characters, but requires multiple \stackunders and some bracing.

The appearance is a bit different, however "the same elements of each reaction are below each other", as requested by the OP. While there are no other alignments that apply one may observe that the spacing on any given line is "correct".

\documentclass[border=10 pt]{standalone}
\usepackage{mathtools,setspace,stackengine}
\begin{document}
\renewcommand\stackalignment{l}
\renewcommand\stacktype{L}
\setstackgap{L}{1.4\baselineskip}
\stackMath
$\displaystyle
^{12}\mathrm{C}+^{1}\mathrm{H}\longrightarrow             \stackunder{^{13}\mathrm{N}+\gamma}{
 ^{13}\mathrm{N}\longrightarrow                        \stackunder{^{13}\mathrm{C}+e^{+}+\nu}{
  ^{13}\mathrm{C}+^{\mathrm{l}}\mathrm{H}\longrightarrow  \stackunder{^{14}\mathrm{N}+\gamma}{
   ^{14}\mathrm{N}+^{\mathrm{l}}\mathrm{H}\longrightarrow \stackunder{^{15}\mathrm{O}+\gamma}{
    ^{15}\mathrm{O}\longrightarrow                     \stackunder{^{15}\mathrm{N}+e^{+}+\nu}{
     ^{15}\mathrm{N}+^{1}\mathrm{H}\longrightarrow           ^{12}\mathrm{C}+^{4}\mathrm{He}
}}}}}
$
\end{document}

I suggest to align your entries like the way I did (using Tab) to make it easier to read and modify if needed. I have used \newcommand to create a way for you to enter the data of a row and every cell in the array is encapsulated by \mathrm{...}. Also, I changed the 23 cs you had in the array to *{23}{c}.

\documentclass[border=10 pt]{standalone}
\usepackage{mathtools,setspace}
% Usage:
% \data{operand-1}{operator-1}{operand-2}{operator-2}{operand-3}{operator-3}{operand-4}
% Simple example:
% \data{a}{+}{b}{-}{c}{*}{d}
\newcommand{\data}[7]{%
    \ensuremath{%
        \mathrm{#1} & \mathrm{#2} & \mathrm{#3} & \mathrm{#4} & \mathrm{#5} & \mathrm{#6} & \mathrm{#7}
    }%
}%
\begin{document}
$\displaystyle
{\setstretch{1.5}
\begin{array}{*{23}{c}} 
                    \data{^{12} C}{+}               {^{1} H}    {\longrightarrow}   {^{13} N}   {+}{\gamma} \\
&&&&                \data{^{13} N}{\longrightarrow} {^{13} C}   {+}                 {e^{+}}     {+}{\nu}    \\
&&&&&&              \data{^{13} C}{+}               {^{1} H}    {\longrightarrow}   {^{14} N}   {+}{\gamma} \\
&&&&&&&&&&          \data{^{14} N}{+}               {^{1} H}    {\longrightarrow}   {^{15} O}   {+}{\gamma} \\
&&&&&&&&&&&&&&      \data{^{15} O}{\longrightarrow} {^{15} N}   {+}                 {e^{+}}     {+}{\nu}    \\
&&&&&&&&&&&&&&&&    \data{^{15} N}{+}               {^{1} H}    {\longrightarrow}   {^{12} C}   {+}{^{4} He}\\
\end{array}}
$
\end{document}