iOS (iPhone) "Other" storage category is huge
As Steve Chambers commented - this was the ONLY thing that worked:

Backup phone on itunes to your local computer (not iCloud).
Wipe/Reset phone completely.  Delete all data and settings.
When it comes back up, plug into your computer again (or just leave it plugged into your commputer the entire time).
In iTunes, click restore from backup (the one you just made locally on your computer).

It will reload all your apps and settings to nearly the exact same (I had some things that randomly didn't restore properly and had to handle myself).  You'll also have to login again to some apps, but others you won't.
It also takes a long time, and is generally a pain.  But, given that this problem is all over the internet including on apple forums with no response or help from the company (typical), I will depend on this in the future too!