David:
I have a synthetic hone system that finishes with the Suehiro G20. I also hone on naturals, mostly J-Nats.
It's been my experience that the G20 is a very fast finisher as long as you take a shave ready edge to it. To me this means the equivalent of a maxed out 8K or better edge, and a comfortable shave but lacking as close as possible.
On the razors that I have honed on this finisher I will max out the 8K level and then do about ten laps on the G20, strop and test to see if the edge is where I like it. If I find that I would like a bit more then another ten laps and again test. I have never found to date a reason to do more than 30 laps on this finisher. Usually less is better if my early work is done well.

I don't hone on a Thuri or a Coti but if I did I would try the G20 after the Coti first, see what kind of results you get. If you are happy then your Thuri would not be useful in that progression.

I don't think that I would use the G20 in my natural progression because I get what I want in my edges on naturals only, but I know others that do mix naturals with the G20 with results they like.
I have shaved off of very good Coti edges and I believe that if I could attain that level of shave comfort I would choose to go right to the G20 from the Coti, no Thuri bridge necessary.
Part of the fun of having these stones is the ability to use them in different ways and experiment to see what edges you like the best. I hope your stone arrives soon so that the fun can begin.