Adobe fonts and Accessible PDFs

But are you saying the Accessibility set of tools in Acrobat are designed to fail? 

When I look at Adobe's help page regarding creation of accessible PDFs from InDesign, I see that it suggests that the majority of the accessibility work should be done in InDesign, not Acrobat. And the small number of things it tells you to do in Acrobat? You can do 'em all from InDesign, except "run PDF accessibility checker."  So that's why I was asking you "what are you doing in Acrobat?" Because the fast way to get around the problem of "my fonts unembed when I save my remediated PDF in Acrobat" is "don't remediate in Acrobat."

Doing remediation in Acrobat is a form of PDF editing, right? At many times over the years, there have been occasional weird consequences that come from trying to edit PDFs. That's why we usually say, around here in the InDesign forum, that it's best to treat the PDF as a final container format, not something easily editable. To be direct, I suspect you've uncovered a bug in Acrobat where font embedding of Adobe fonts is damaged by remediation. Having encountered numerous bugs with Acrobat's accessibility remediation tools myself, I'm not terribly surprised. Those bugs, when reported to the Acrobat team, usually get fixed. That's the opposite of "designed to fail" - the Acrobat devs are clearly trying to make it work. 

But why would you need to do accesibility remediation in Acrobat? It does matter, a lot, where that work happens. If you build your InDesign documents to be accessible, then you most likely don't need to remediate afterwards. You just export your tagged Interactive PDF, drop it in your CommonLook or your PAC 2024 or whatever.  It tells you if it finds any issues. If it does, then you hop back into InDesign and fix it there. If that's your workflow, then you never need to remediate and resave in Acrobat, and you never trigger this font-unembedding bug you've found in Acrobat.

On the other hand, maybe you have some regulatory need to remediate in Acrobat. Or you don't have any say in your employer's workflow, so there's no point in your saying "But doing post-export remediation in Acrobat is deprecated!" to your manager, because no one will lisen, or something along those lines. If so, consider posting your question in the Acrobat forum, and pursuing a bug report for Acrobat. If typical print and interactive PDFs exported from InDesign have successfully embedded fonts, and the fonts only drop when you're trying to remediate in Acrobat, then the problem is clearly to be found with how Acrobat reads fonts embedded from Adobe Fonts when resaving.