Continuity is really interesting since it really highlights just how far we've gotten from the original promise of seamless integration between Apple products. For starters, its taken years to get to this feature (with incredibly confusing pitstops in the middle, such as the existence of a feature called Airdrop on Mac AND iPhone that didn't work between them....).
All in all though, continuity is another disappointment to me. We are still stuck in app-land. Handover is app-to-app. So when I'm listening to a podcast in Safari, I won't be able to handover to Overcast on my phone. I actually have to wait for marco arment to write a desktop podcast player, and worse, I have to use it, to get this basic functionality. Again, an aggressive misunderstanding of what it means to have true continuity: reading the same data on two devices, not using the same app on both. But Apple thinks wallet first, and "App" is their ecosystem's bread and butter, and so this way it will stay.
This is particularly depressing since I remember, 10 years ago, having (what I believe to be) much more impressive interconnectivity between my Mac and my Nokia phone through Saling clicker. When someone called me, my music would pause. I could control my mac with a really cool remote. Instead, we seem to be living in a parody of that world where now if someone calls me 10 devices in my house go off since Apple "connected" them all together to ring simultaneously, its maddening.
Apple (and others) have clearly double downed on an app-centric metaphor. This is obvious from App Store and home screen only showing apps--interestingly they did add recent people to the multitasking menu). They're also struggling with trying to reinvent documents (and haven't done much). Services/Workflow-centric and people-centric are the other approaches I've seen attempted[1] with only minor success. I don't blame Apple, none of the other approaches have gained any traction and Apple has really streamlined app centric management: centralized finding apps, purchasing, encapsulating installing/uninstalling, updating, etc. All of those things were pretty inconsistent and terrible before OSX/iOS.
I also wished they refined more of the Bluetooth integration that OSX had 10 years ago (sending/receiving txt messages, using it as a remote, etc). Sadly, it only now seems to be coming back and in an iPhone-specific way. But other than the iPhone tie-in, what would be different? If you had your phone Bluetooth paired to multiple devices, wouldn't they all buzz? If you don't want them to buzz, can't you just disconnect them (as you would un-pair bluetooth)? If you had a remote tethered to multiple devices, would "Play" trigger them all at once?
I dunno. Continuity seemed to me to be novel and useful when it was presented. Healthkit is new, and not really "just a refinement".