I recommend to not bother with the spinny thing. I have one, and while I did not throw it out while cleaning the garage this weekend, I just take my chain off, soak it in degreaser, scrub it all over with a cloth while sudsy, clean that all off with water, and relube after reinstalling (make sure the degreaser is fully removed before relubing).

edit: And, as TS said above, can then clean the chainrings, cassette, derailer pulleys... for the most part I can clean those thoroughly with a dry paper towel, no need for solvent at all.