I believe they are semi tubular flat head rivets -- at least mine are. I would be concerned using full tubular or hollow rivets as they could collapse if the door check was pulled hard against them.
One downside to working with rivets is they need a proper setting tool for install. Most of those "easy to use tools" are expensive and large for that size rivet. The distance between the door jamb and the point were the rivet end has to be collapsed is very limited. I don't know if a regular lever operated setting tool is that small or would fit. The other alternative is the hammer or hand set clincher tool but that would be hard to use from the bottom.
The old cut off rivet is just sitting in my 56 and gravity holds it in. As an alternative, if I were going to do it again I would use the low profile shoulder screw with threads ground off to fit flush with the nut-- even though it has a slightly wider head than the rivet. The socket hole could be filled, and if painted I doubt anyone would know it wasn't a rivet. A pan head threaded rivet with no socket hole to worry about filling would also work but its head is wider than the shoulder screw and the nut size would be larger too. The space on the bottom for a nut is limited due to the slot width in the door.
With either, the bottom leg of the rivet mount could be drilled to accept the knurl of a capture nut or just grind that knurled portion off the nut to leave a round thin profile on the bottom. It could be kept from unscrewing with locktite and could still be removed with relative ease if the door ever had to come off again. You could also use a thin round speed nut or push nut to hold the screw.
Here are some illustrations of the screw items. There are other sizes and lengths available so the proper size would have to be determined. Restoration Supply has the threaded rivet as well as semi tubulars. McMaster has the shoulder screws, capture nuts and also semi tubular rivets.
EDIT: After some thought I wonder if a hand set clincher tool could be adapted and mounted or used in a large C clamp to set the bottom of the rivet. That might be small enough for the space and the opposite jaw of the C clamp would press on the top of the rivet to hold it so the tool could collapse the bottom. Another possibility is maybe to modify the end of a C clamp screw to not need the tool.
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