For any colonial friends, it would sound very natural indeed, because of
Bring me your tired, your poor, your ...
huddled masses, et cetera.
It is the same construction, omitting "persons," and implying group.
Note too that "wounded" is also very commonly used in this way. "Bring me your wounded," "Bring the wounded here..."
{Thanks to action movies that such phrases exist outside a limited circle!}
Chiastic, regarding your question I'm surprised it sounds unusual to you - reads totally normally to me in BrE.
(The fact that "sick" can also mean "vomit", or indeed be an adjective, "I'm sick", and nowadays most commonly means "well-executed skateboard move" - so what? English is incredibly ambiguous at every turn, it's not even worth mentioning when you see a sentence which has a humorous alternate meaning, you know?)
(For example, the immediately preceding sentence happens to have a humorous Monty-Python interpretation ... sending word to the "country": you can picture Eric Idle addressing a large green field and trees.)