By an odd fluke, my local hardware store had a box of brass rivets, forgotten by the original customer. The rivets have a 12 mm flat head and a 4 mm diameter shaft that is 9 mm long. The shafts are solid to the middle and then hollow.
As some of the respondents to my inquiry noted: the shaft of the rivets do all the "work", the heads, besides holding the rivet in place, are mostly ornamental that is why the root of the shaft is solid
Anyway, once I got them home I had to try replacing a rivet. So I drilled out an old one and placed new one in its hole.
Rivet replacement is not difficult. It can even be done without a rivet setter as follows:
cut the hollow shaft radially so that it will bend over easily hammer a cone (I used a punch) into the opening to bend the "petals" evenly use a ball peen hammer to finish the bend and tighten the rivet use a ball peen hammer to contour the head over the leather at all times support the other side of the rivet with a block of hardwood or piece of metal
This takes time and patience, but can look quite nice afterwards.
The technique gets a lot more complicated when the leather on the saddle has been completely removed. I'll report on that when I get to it.