You have num:2, so subsubsections are not TOC'ed, so they don't get
   the alternative. If you set it to 3, all should work.

That is what we have been discussing. There are situations where you
do not want a headline to appear in TOC, but still want the ALT_TITLE
used. It is now possible in org-mode, but there was some bug because
of which it did not always work.

I figured I was missing something, but I have a question (which means
I'm still missing something :-) : what are ALT_TITLEs good for if not
for the TOC?

I think some styles put them in the header, regardless of TOC.