I don't think the UK was asking for a dragnet.
The normal practice is if somebody is a target of interest, and the proper court order has been issued by a judge, then the authorities have the expectation that they can ask for access to private data, or have the ability to put in a 'tap'.
Under Apple's ADP system, Apple are unable to give that access - only the account holder can - and obviously they may not want to - and asking them will obviously alert them they are under surveillance - if that was the original aim.
So the talk about a 'back-door' in the Apple product for the UK government is a bit misleading - in the sense they are not asking for direct access that avoids having to ask Apple - they are just asking Apple to build functionality so Apple can fulfill such requests.
ie If the government get's a 'search warrant' Apple has the ability to comply.
Ironically if GCHQ did have a backdoor without needing to ask Apple then they could do much more dragnet stuff.