I followed a similar process, and even received replacement AirPods from Apple. Same results: with a clean or restored from backup iPhone 6S, AirPods work perfectly. After re-pairing my Apple Watch (original 38mm sport model), the audio glitches and AirPod disconnects during calls start happening again. Doesn't seem to matter if I run phone and watch setup as new or restored from backup.
I can even provoke the symptom by going to airplane mode, re-enabling Bluetooth, and listening to the locally stored voicemails in the phone app. It is less likely to happen with just one AirPod.
The symptoms are definitely pointing towards my iPhone 6S not handling more than one or two Bluetooth devices connected simultaneously. It is as if some condition accumulates over time and the iPhone 6S Bluetooth radio resets, dumping everyone off (including the Watch). Could be a problem in the Bluetooth radio chipset, or the software managing it.
I have access to an iPhone 7 Plus this week and may give that a try. Problem is the disconnects are so random it could happen early and often, or after several successful calls.