Sum-class personalised operators

Question

Following Bourbaki, Algebra, I wanted to describe the 'composition of an ordered sequence' of elements of a magma with an operation that is neither + nor \cdot so with a different big operator like \sum. Herewith a scan of the bit. Please ignore the fact that it is in line: what I wish to have should be acting like \sum.

I have seen this similar post and this other post but neither respond precisely to my problem.

1. I wish I could have a command like \bigop[1] in which I can choose the symbol wanted (e.g. Bourbaki would use \bigop{\top} and \bigop{\bot}.

2. The symbol should be vertically centred, like \sum and \prod are.

I have tried the following, but it seems to me it is a not particularly elegant solution. I hope someone has a much more elegant one.

MWE

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\newcommand{\bigop}[1]{\mathop{\raisebox{-3pt}{$#1$}}
\begin{document}
[
\bigop{*}_{a \in A} x_a
]
\end{document}