Align equation while centering symbol

Like this?

I only slightly change/simplify your MWE ...

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{array}

\begin{document}
\[\setlength\arraycolsep{3pt}
  \renewcommand\arraystretch{1.3}
    \begin{array}{>{\displaystyle}cl}
\bigwedge\limits_{a'\in\mathcal{A}} & \text{some text goes here}\\
\bigwedge\limits_{a':a'>a}          & \text{and here} \\
\bigwedge\limits_{a':a'<a}          & \text{and here}
    \end{array}
\]
\end{document}

Edit: with added \displaystyle the \bigwedge is thew same as in simple equation. For adding this option the package array is added into preamble of MWE.

Use the \smashoperator command, from \mathtools. Also, \colon has a better horizontal spacing than :

\documentclass[11pt]{article}
\usepackage{mathtools}

\begin{document}

\begin{align*}
    & \smashoperator{ \bigwedge_{a' ∈ \mathcal{A}}} \text{some text goes here} \\
    & \smashoperator{\bigwedge_{a'\colon a'>a}} \text{and here} \\
    & \smashoperator{⋀ _{a'\colon a'<a}} \text{and here}
\end{align*}

\end{document}