Has anybody tried just compiling the list of languages/scripts where the new fonts don't work, and simply disabling the fonts for them? Because that would be a much better solution than continuing to revert this change in successive release branches.

(From the technical side, that can be trivially done using the :lang() selector, supported on all browsers other than IE<6: http://caniuse.com/#feat=css-sel2. We already use this technique to disable link underlines in some languages.)

Has anybody tried just compiling the list of languages/scripts where the new fonts don't work, and simply disabling the fonts for them?

Well, that wasn't done because no language testing was ever done in the first place for this font change. Can't we instead whitelist the languages which have been verified to work?
As a start, we could blacklist all non-Latin script languages.

This patch might be related: https://gerrit.wikimedia.org/r/#/c/200940/

We're not getting this done in time for the MediaWiki 1.25.x tarballs. It's not a regression from the releases in the 1.24.x series, so, pulling from there too.

Change 216028 had a related patch set uploaded (by Isarra):
Make headers use the same fonts as the rest of the content

https://gerrit.wikimedia.org/r/216028

Language testing was done for this typography update in 2012

Please see this page where we were recording issues for non latin scripts:

https://www.mediawiki.org/wiki/Typography_refresh/Non-Latin_scripts
The old type size in non-Latin scripts was 0.8em (12.8px). This squashes glyphs and superscripts significantly along with the type being too small to read.
Scripts examined were Urdu, Marathi, Bahasa Melayu, Chinese, Korean, and Navajo. The body copy type size increase will improve readability for most scripts. Specifically for Navajo, an override will be provided because character pairs render strangely in Helvetica.

Inline CSS guidance can be provided to ensure that languages make overrides on a case-by-case basis as needed. Please reach us/comment on the Talk page if you primarily use a non-Latin script and encounter significant problems. We want to get this list together.

It would be nice to have a release that I'm not ashamed of promoting to non-English users. 1.23 is LTS but it's getting old.

Change 216028 abandoned by Isarra:
Make headers use the same fonts as the rest of the content

https://gerrit.wikimedia.org/r/216028

Patch provided at https://gerrit.wikimedia.org/r/#/c/217707/ and released in 1.28.10 https://phabricator.wikimedia.org/diffusion/SVEC/browse/wmf%252F1.28.0-wmf.10/ featuring generic font-family: sans-serif specifically for Korean, Hebrew and Japanese fixed this issue stated in the task description.
If there's any language mistakenly left behind please reopen and comment or preferably file a new task.

featuring generic font-family: sans-serif specifically for Korean, Hebrew and Japanese fixed this issue stated in the task description.

Does not seem so to me.