Latin Modern Math doesn't have a bold font, but does have glyphs for Unicode block Mathematical Alphanumeric Symbols. Can \boldmath output them?

Question

It is known that Latin Modern Math does not have a standalone bold font. If you load Latin Modern Math with package unicode-math, and try to use \boldmath, the fontshape with be substituted with the regular font. A minimum working example comparing \boldmath to the unicode-math commands \symbfit & \symbfup, which output bold Latin Modern Math as desired, is as follows:

\documentclass{article}
\usepackage{unicode-math}
\setmathfont{Latin Modern Math}
\begin{document}
{\boldmath $a~A~\omega~\Omega$} \par
$\symbfit{a~A~\omega}~\symbfup{\Omega}$
\end{document}

The compiler reports:

The bold Latin Modern Math glyphs come from the Unicode block Mathematical Alphanumeric Symbols, as shown below (taken from the unicode-math documentation):

My question is:

Can I redefine \boldmath to make it behave as desired by outputting the bold Unicode characters?

---

To answer in advance "Why must you use \boldmath?", it is because I have been using \boldmath a lot in my documents before starting to use unicode-math. My main usages are:

1. Bold math in section titles, but implemented in such a way that does not interfere with the formatting of Table of Contents.

\documentclass{article}
\makeatletter
\g@addto@macro\bfseries{\boldmath}
\makeatother
\begin{document}
\tableofcontents
\section{Bold math in section titles $a~A~\omega~\Omega$}
\subsection{But regular font in ToC $a~A~\omega~\Omega$}
\end{document}

2. Make all characters including operators bold, instead of just the letters.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\boldmath $2\hat{a} + \tilde{b} = \sqrt{\dot{c}}$ \
\unboldmath $2\hat{a} + \tilde{b} = \sqrt{\dot{c}}$
\end{document}

P.S.

1. I also plan to to apply the solution to \bm if possible.
2. The reason why I started using unicode-math is that I want to import some glyphs from other fonts, like integrals from XITS Math, \mathscr from STIX Two Math, \mathbb from TeX Gyre Termes Math, etc. I can achieve that in some other ways, but I don't want to give up unicode-math without a fight.

Answer 1

I don't think that it would be a good idea. Only a rather small subset of the mathematic has bold versions. It is better to use FakeBold:

\documentclass{article}
\usepackage{amsmath}
\usepackage{unicode-math}
\setmathfont{Latin Modern Math}
\setmathfont{Latin Modern Math}[version=bold,FakeBold={3}]
\begin{document}
$a~A~\omega~\Omega$ $\hat f(t) = \int_0^t \hat f'(x)\, dx$
{\boldmath $a~A~\omega~\Omega$ $\hat f(t) = \int_0^t \hat f'(x), dx$}
\end{document}

Answer 2

You could use a font that has both weights.

New Computer Modern Math is another derivative of computer modern, so very similar to latin modern but more extensive and with two weights. Its Book weight isn't very bold but is noticeably darker than the Regular:

\documentclass{article}
\usepackage{unicode-math}
\setmathfont{NewCMMath-Regular.otf}
\setmathfont[version=bold]{NewCMMath-Book.otf}
\begin{document}
[y=\sum f(x)]
\boldmath
[y=\sum f(x)]
\end{document}