From https://wikiwho-flower.wmcloud.org/ (maintainer-only), I have deduced that very large and revision-rich articles like Barack Obama cannot feasibly be processed from scratch – that is, to process the entire revision history to create a new pickle storage file when one did not previously exist. It will always time out. This is why we used XML files for the initial import, both to reduce the work the algorithm has to do, and to avoid having to query the action API en masse. For pages with an existing pickle, this is not a problem, as we only need to append to the storage as new revisions are made, and don't have to reprocess the entire revision history.
In testing T414075, I decided the delete the pickle file for [[Barack Obama]], and so now that page is indefinitely unusable in WikiWho.
To make matters worse, new requests come in and hog the queue when trying to rebuild the pickle file, and I think that's more or less our issue with the queue going offline. The initial request will be under the worker_user queue, and you can search for it in Flower. That will eventually time out, then it gets moved to worker_long, where it will time out again after an hour or so.
The new global API limitations at Wikimedia also may be at play here (though Cloud Services is meant to be exempt). At any rate, it's clear we need to do something. Quite a few other articles are unavailable in WikiWho for the same reason. It stands to reason for Wikipedias that high-revision mainspace articles are probably of great interest, so ideally Who-Wrote-That, Education-Program-Dashboard, XTools and other integrations will be able to support said articles.