[Eliot Ness]

......Now what I have to understand is how it looks (how it goes) inside the wood.
And also, what is the tool that is used to put them in?
Plus, what size is needed......

Like Paul mentioned, digging out those split rivets often digs out too much wood to use new ones, but for reference I'm also curious what tool is used and the correct size.  Looking at the picture below you can see how some tools actually curled the ends of the split rivets.  I suspect the tool Zenith used did the same while burying the ends into the wood, thus making removal difficult.

   

As I mentioned earlier I think using a blind brass screw (round head with no slot) is the way to go, and you could even just drill out the center post that actually holds the handle in place and replace it with a blind screw.  You'd have to drill a small hole under the handle so a nut and washer could hold it in place, but that way you're not digging out the 4 original rivets.  Unfortunately I've not been able to find a blind brass screw of the correct size.

What some guys use are regular screws they paint a brass color, but I'd much rather find some blind brass screws of the correct size.

Edit: doing more searching it looks like Restoration Supply Company sells threaded brass rivets (essentially a round head with no slot) on page 38 of their catalog along with split rivets and tool on page 39:

http://www.restorationstuff.com/pdf/Rest...ompany.pdf

https://www.restorationstuff.com/ecommer...&cPath=200

I'm not sure if 1" will be long enough for the center that holds the handle but it should be for the sides and I just ordered 4 each of the 1"  147, 148, and 602 to see how well they'll work.