- In the comments it was pointed out that my original approach (Approach 2) is maybe not always legal w.r.t. the license of the font, so I added "Approach 1".
Approach 1: Convert PDF to images and include them in a new PDF
- This should be legal with all fonts that are embeddable into a PDF file.
pdfpages can be used to include the generated image files in a new PDF file again, keeping the same optics.
Approach 2: Patching the PDF file (when the license allows it; e.g. SIL Open Font License)
Thanks to the comment by Ulrike Fischer I was able to find the relevant strings when compiling in uncompressed mode:
\edef\pdfcompresslevel{\pdfvariable compresslevel}
\pdfcompresslevel=0
\documentclass[a4paper]{article}
\usepackage{fontspec}
\begin{document}
\setmainfont[]{XITS-Regular.otf}
Example
\end{document}
Then an easy search and replace does the trick.
For a generic solution I will implement a decompressor and recompressor or compile in uncompressed mode and include the uncompressed PDF file using pdfpages and recompress in the second run.
Here is how the patched PDF file looks:

/YISSQH+LMRoman10-Regular. But they don't show up in the Adobe dialog. – Ulrike Fischer Nov 15 '21 at 10:59