Compiling
\documentclass{article}
\usepackage{unicode-math}
\begin{document}
$a\mathbb{1}$
\end{document}
with lualatex yields

(The a is there just for the sake of comparison.)
Similarly, running pdflatex on
\documentclass{article}
\usepackage{bbold}
\begin{document}
$a\mathbb{1}$
\end{document}

The two results look quite identical, so I don't really understand your claim that
I would prefer NOT to use the bbol, mathbbold, bbm, ... packages. The only package I've encountered so far that provides a nice rendering of the character I'm looking for is unicode-math.
In fact, the standard math font with lualatex and xelatex is Latin Modern Math, and its documentation states explicitly
the double struck script is excerpted from Alan Jeffrey's bbold font
My guess is that your problem with loading bbold is that it overwrites the definition of \mathbb from amssymb. This can be easily circumvented by adapting the code of bbold.sty (it's just six lines of code)
\documentclass{article}
\usepackage{amsmath,amssymb}
\DeclareMathAlphabet{\mathbbold}{U}{bbold}{m}{n}
\newcommand*{\boldone}{\mathbbold{1}}
\begin{document}
$\mathbb{R}$ % that's still from amssymb
$\mathbbold{R}$ % that's from bbold
$\boldone$ % that's from bbold
\end{document}

Note, however, that once you use the symbol in a formula this solution will allocate a (possibly precious) math family; not an issue if you are not using lots of other fonts, but sometimes packages do things without you noticing. If you need only the 1, you could alternatively define
\newcommand*{\boldone}{\text{\usefont{U}{bbold}{m}{n}1}}
pdftex, thenbboldshould be in fact the correct choice, so then please explain why you would not prefer using it. If it's about the redefinition of\mathbbfromamssymb: that can be easily taken care of. – campa Oct 14 '20 at 12:24\usepackage[default]{fontsetup}to switch to New Computer Modern. – Davislor Oct 14 '20 at 17:30unicode-mathwithamssymb. It doesn’t cause an error, butamssymbwill have no effect. – Davislor Oct 14 '20 at 17:33