"- The ring handles of the "pocket saw" are way to big without any benefits that I can see."
That is so that you can something else -- like a piece of a stick -- through the rings to make a better handle.
ah... right. Yes, agreed, if you have only the rings those rings will have to be of that size to be of any use.
A loop of cord does exactly the same thing, but packs much smaller and is easier and more versatile to work with. If you use your hands / fingers (no handle) then a loop is also much more comfortable.
No big deal - replacing those rings with some cord takes about 2 minutes.
Note that the BCB version I linked to above use beefy and comfortable straps, not rings...