I've been experiencing this issue since before iOS 9. Every update, I cross my fingers and hope it got silently resolved, but no luck so far. I can't pinpoint a time in iOS 8's lifespan that the issue started, but it has been annoying me for many months.
I have 8 gmail, 1 yahoo and 1 iCloud account on my phone. Yahoo and iCloud never have any issues. The gmail accounts range from standard gmail.com accounts to Google for Domains accounts for various domain names. Sometimes it's a single account. Sometimes multiple accounts start causing issues. It's always a random account that starts causing issues and it never resolves itself over time. The only way I've found to alleviate the problem is the same as you, delete and re-add the account and things will be fine for a few times.
I've tried a lot of different things. I've changed most of my accounts to fetch manually and only have the 2 important accounts set to fetch automatically. I've tried changing the interval of these fetches and that hasn't done anything to help. I've also changed all of my passwords for each of the accounts. Deleted and re-added them all and crossed my fingers. A few days later, I get the same message again. I've gone through the Google captcha page for each account which is supposed to "unlock" the account. That doesn't work either.
I'm at a loss on what else can be done. I feel like the only possibly solution I haven't tried yet is a complete wipe and reset and setup the phone as "new" instead of restoring from a backup. I have restored from backup multiple times while having my iPhone 6 Plus replaced by Apple when I was having issues as well as when I upgraded to the iPhone 6S Plus recently. I would have reset the phone to new a long time ago now, but Apple doesn't allow for any sort of selective restoring so I would lose all the health data that has been collected over the past 6 months from my Apple Watch and I would really prefer not to do that.
Any other assistance on this matter would be greatly appreciated. I filed a report on Apple's bug reporting radar.