Disabled iPhone, screen broken, no access from pre-synced computer!?

How do I access my disabled iPhone 5s with a broken screen, on my pre-synced computer - which by that logic should always have access to the device!

I've tried recovery-mode-to-update but that doesn't do the trick, since iTunes will always prompt me to type in the passcode on my phone. Which I do have but am unable to do so.

Surely Apple Support must provide some sort of remote access service, provided all the necessary information, enabling me to gain access to my device, and in this case, back it up?

Thanks.