I have an apparent solution which I have tested on on two phones with ios10.3.2 installed. Both are iPhone 6.
An online search led me to check out the Accessibility section of the Settings on the iPhone. The applicable section there which was named INCOMING CALLS is now CALL AUDIO routing! Why is this in Accessibility?
This is what was happening:
On the iPhones with Bluetooth earbuds connected, an incoming call could be heard on some calls on the speaker of the phone but not in the earbuds. A window would appear in which the bluetooth earbuds would be shown as a choice. Tapping that did switch the audio to the earbuds. On other incoming calls the audio could be heard both on the phone's speaker and the earbuds and tapping the bluetooth icon would switch to the earbuds only. The microphone on the earbuds was not turned on on every call either. This is inconsistent and therefore undesirable! This behaviour was the same with two phones and two different pairs of bluetooth earbuds!
This suggests at least one error in the ios software. But there is another as well.
I went into Settings ...General...Accessibility and scrolled down to Call Audio Routing
I tapped that and switched from Automatic (which was obviously manual as described above) to Bluetooth Headset.
A test call resulted in the call going only to the earbuds on on each of the phones. The microphone on the earbuds working too! Then I made another call with the earbuds disconnected to determine what would happen. The call went to the speaker and microphone on the phone.
Conclusion? Automatic setting is NOT automatic. Bluetooth Headset setting appears to provide the desired automatic functionality if no bluetooth headset is connected. This could be as simple as those two options being labelled incorrectly.
Apple, if you read these postings, please fix this and the inconsistent results mentioned above in the next ios update.