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They just volunteer.
Isn’t Jerusalem itself a city? Or is it more similar to New York City, with the area sectioned into boroughs? >

Jerusalem is the city itself. "City" of David isnt an actual city but an archeological site where the city was so it got "city" as a nickname.
Inside the modern city there is an area called "city of David" which was where ancient Jerusalem stood. Today it's a half archeological site half museum.
The oldest structure is dated to around 1200-1300 BC >


Today it's a half archeological site half museum.

...and all tourist trap. >

This is the best tl;dr I could make, original reduced by 77%. (I'm a bot)


A rare gold bead made by hand in Jerusalem some 1,600 years ago was recently discovered by an 18-year-old volunteer in the City of David, not far from the Old City, the Israel Antiquities Authority said Wednesday.
"Throughout all my years in archaeology, I have found gold perhaps once or twice, so to find gold jewelry is something very, very special," said Dr. Amir Golani, an ancient jewelry expert at the IAA. He noted that the bead was probably just a small part of a necklace or bracelet that included additional beads: "Whoever could afford a piece like this made from gold was a wealthy person of means."
To date, only a few dozen gold beads have been found in Israel, including one that was found two years ago by a 9-year-old while sifting earth from the Temple Mount.


Extended Summary | FAQ | Feedback | Top keywords: bead^#1 gold^#2 IAA^#3 found^#4 structure^#5 >

I always wonder do the people who find antiques get paid or does the government just confiscate them? >

He didn't just stumble upon it, there's a site where visitors can help by filtering artifacts through mud. >

I assume those visitors do get paid right or do they just volunteer for it?