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Anyone have experience with these type of fasteners? I'm assuming I would have to drill them. But what would I use to go back together?

Another picture. I could drill them out and replace with rivets however they are not flush on both sides which would block the matchbox from vending

I believe those are called Leather Rivets or Solid Rivets. They make them in brass, steel, aluminum and stainless steel.Brass would be your best bet as it is ductile and flattens on the back end easily.

Here's a source:

https://www.rivetsonline.com/rivets/solid-rivets