I'm having a similar issue.
I'm wondering if support for downloading data from hosted tables (rather than hosted feature layers with actual geometry) was dropped or screwed up in a recent update? I have a dashboard more than a year old that consumes a hosted feature table with no spatial info, and the download option was available and working all that time. Recently, I noticed the download button stopped working and would give a 505 error instead. Now (not sure if related or unrelated) my hosted table went corrupt or something and I had to re-upload it entirely, but no matter what I do I can't get that download toggle to appear again, despite having the Extract capability enabled in the item settings.
EDIT: Just realized this post was in regard to ArcGIS Enterprise 11 -- my issue is on ArcGIS Online
-Aaron