Reaper does store some things in the registry but not much. Rearoute dll seems to be registered with the system. I wonder if Rearoute can even be used from a non-installed USB key launch? I don't know. Try it on a system where Reaper was never installed.
I don't think Justin ever claimed nothing was written to the registry. There's not much there though beyond program paths and some other things that Windows probably does on it's own.
Compared to most windows programs it might as well be nothing.
I know Rearoute is registered because it shows up in Cubase, and it's rewire outputs, which also show up.