What are your everyday knives and japanese knives? I find that jnats are only effective on knives that are very thin behind the edge and that can hold this thin edge, otherwise synths are much better.
For easy to sharpen carbon steels on double bevel knives, i like to use (for fun) a progression starting with Numata (from JNS) to set the bevel. JNS's Numata can be hard to come by, so you could look for the right omura or use a synth.
Next in the progression could be an Aizu (or Ikarashi then Aizu). This will give you plenty of tooth, but it took me a few months to start liking Aizu as it was too hard for my taste in the beginning. Another option after JNS' Numata would be a nice muddy aoto if you favor softer stones.
If deburring is a problem, i would add this Natsuya after Numata:
http://www.japanesenaturalstones.com/natsuya/ or the muddy aoto if you hadn't used it already. Both are softer and have better feedback than Ikarashi/Aizu. This Natsuya is not fast and it is much softer than other Natsuyas i have used before. Aotos can be much faster though.
Aizu leaves a toothy edge that is great as a final stone for gyutos, but it is from here on that i find that some of my best results happened. There are a lot of options. My first edge that made me love jnats was with a Tsushima on a Kochi kurouchi gyuto. Tsushima is relatively cheap, not a fast stone and it's not really popular on KKF, but it did the trick for me. Instead of Tsushima, a fast suita is nice, but usually more expensive if you are comparing similarly sized stones.
The point to keep in mind is that you don't really need jnats as the right synths are more efficient and can leave great edges. Jnats are great, though, to make one strive for more delicate edges. They can help you to achieve sharper edges? I think so, but i feel one's ability to sharpen is really what makes the magic happen.
I know very little about polishing, so i'll leave that matter to others.
