I can't speak for XeTeX, although I am very sure that both LuaTeX and XeTeX can load metafont fonts. So the answer to your question is "yes". But if there is a Type1 or OpenType alternative, I'd always go for that.
LuaTeX loads cmr10 by default, not latin modern. Of course one could change that, but the idea is to get as much portability between the different engines as possible. So if you run a document through PDFTeX and through LuaTeX, both results should be the same.
The user is supposed to use the fonts he or she wants. Computer Modern und thus Latin Modern has the huge advantage to be a) free b) a big set of fonts (italic, monospaced, sans serif,...) and distributed with all ancient TeX systems. If the distributions chose another font as the default font, nothing much will be gained as there is not really much to do more than:
\documentclass{article}
\usepackage{fontspec}
\setmainfont{Linux Libertine O}
\begin{document}
Hello world.
\end{document}
While in principle I like your idea, I think it is extremely unlikely to ever happen, so I would not bother to think too much about it.
XeTeX simply doesn't support the \pdfglyphtounicode primitive, so this route is not available. To understand why, remember that pdfTeX is 8-bit while XeTeX is natively UTF-8. AS such, it's perfectly reasonable for XeTeX to expect fonts to be 'properly' constructed to have glyphs in the appropriately-named slots and therefore to work correctly in the PDF output with no 'trickery'.
Taking the 'what to do' side of things, the problem is most likely with the font. For example, some fonts have the correct glyphs but in the 'wrong' places (e.g. in the Private Use Area). That's really not a 'TeX problem': all that can be done is to talk with the font designers and try to get them to fix it.
Best Answer
Yes, you should be able to, using the PDFX package. But you should realize that there is much more to the PDF/X spec than just embedding fonts. The PDFX package can be had here: http://www.ctan.org/tex-archive/macros/latex/contrib/pdfx/
Otherwise, produce your PDF with embedded fonts, then try and print the PDF to another PDF using OS X's print dialog, and save as PDF/X-3.