To get a clearer picture of terminology, lets move the geography. If someone says 'I live in Melbourne', where do you picture they might live? The CBD? 10km radius around the CBD? I suggest you'd simply think of an area like this.
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Or 'We live in London', and I have conveniently circle 'London Gatwick Airport'

Gosh.
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The co-opting of 'Sydney' to simply refer to some arbitrary (small) distance from the CBD flies in the face of everyday convention and communication.
Sydney's new airport is located in western Sydney (lower case w now and with future growth at Bradfield, capital W) and is appropriately named Western Sydney International; Kingsford Smith airport is in eastern Sydney.
Again, I can only express amazement at the passion against WSI. Its existence and success can only benefit Kingsford Smith, which is chokkers and cannot be expanded either geographically (including into or on Botany Bay

) or via the clock.
And anyone who can find space for a new airport other than where it is is welcome to say so. I guess you could squeeze a runway in if you filled in the Prospect Reservoir, but that would still be west of the A3 curtain

. 'Badgerys's Creek' was announced as the site for
Sydney's second airport (note that phraseology) in 1986, nearly 40 years ago. So even way back then, before Sydney expanded hugely to the west, it was considered the best site; the
only site, really. It was out on hold when they put in the second parallel runway at SYD, but re-started when they realised SYD was reaching full capacity.
Just be thankful the second airport is happening. It will benefit all of Sydney and many outside of it, who can't avoid (price, destination or schedule) having to fly international in or out of there.