I don't agree that smaller stones are not useful. A lot of my stones are smaller than your average bench stone. I am not saying to go out and buy a 4x1 stone but you can go a little under size without issue. A lot of my stones are around 7-7.5x2.5 or so and they were, much, much, much cheaper than a similar stone in the 8x3 range. Sometimes hundreds of dollars cheaper.
Idk about others but I can comfortably sharpen an over sized 240 on a 6x2 stone but that is pushing it. That said, I am more of a scrubber than a single stroke sharpener. For clarity I do agree that larger stones are nicer where you can find and afford them.
As for Takashima, affordable ones are out there from time to time and koppa does not have to mean tiny weird shaped stone or at least not from the vendors I use. Some koppa are oddly shapen but huge or are normal size and normalize shape but thin, thin, thin. Then you have narrow stones that are long or wide stones that are short etc. Maybe these should be labeled as something else though, idk. I just use and collect them.
The later, for me, works well for polishing the blade road.
Anyway, my favorite Takashima (lv 2-2.5) @ 7.5x2.25 cost me $163 about a year ago. I have paid more for less in terms of performance more than once too. A great stone that gets on well with my KS at 250+mm.
That said, the Oouchi that Jon has are fine finishers too. I don't like them, in general, as much as the Takashima I have used but they are still nice finishing stones at a nice price. In the lower cost range I still like a Yaginoshima Asagi around Lv 3-3.5 (JNS) or a nice Ohira Asagi around the same hardness, which have typically proven to me to be faster and more aggressive stones. There are so many good choices out there that won't break the bank. Giving up a wee bit of size, while a personal preference, will net you a stone much cheaper. But again, I'd not suggest a pocket size stone.
Long week, not enough sleep and surely not enough coffee. I hope I didn't ramble too much and realize that is based on my experience with a small sample size of each the stones I mentioned (normally 3-4 of each) and things are not near as consistent as with naturals. Rather, it is what it is. (Yeah, I put a disclaimer in my post--lol.)