I've looked into this some more and I've discovered that you can sync from your Edge 510* to your mobile device (iPhone, iPad etc) using the Wahoo Fitness app, which in turn will sync with Garmin Connect, Strava, RunKeeper, MyFitnessPal etc.
OK, I now have a working Edge 510 and it turns out I was wrong on this. And having read the DC Rainmaker review of the 510, I think I understand why...
In theory, it should be possible. The 510* connects to the iPhone to upload rides using Bluetooth, but the hardware in the unit only allows it to use Bluetooth 2.1, rather than the SMART-compatible Bluetooth 4.0, which is what Wahoo use in their devices. And what everyone else is using. It's becoming the de facto standard.
But Garmin's policy is to deprecate Bluetooth - for the simple reason that they own ANT+ so want to promote that instead. Commercially speaking, this is suicidal. Bluetooth is obviously the way forward. I chose a Garmin because I am a self-confessed Garmin fanboy, but this is the sort of thing that may put even me off choosing a Garmin next time.
Until then, I guess I'll have to rely on tapiriik.
Apart from that, initial impressions of the 510 are very favourable. For anyone who has a 500 already, it's probably not enough of an improvement to be worth considering an upgrade, but I'm a newcomer to this game and it seems to suit my needs very well.
*also the Edge 810, FR220 and FR620