Just because it doesn't empty the tank doesn't make it a "partial flush". Your issue is not that the thing opens and closes quickly: that's how it is supposed to work. As Terry explained, it's using the weight of the full tank of water to create water pressure to make the stuff go down with just a little bit of the water in the tank. You shouldn't need to hold the lever down longer except if you have extra material that needs extra water.
I actually wanted to help address a different statement you made, to the effect that you can't believe Kohler would endorse a defective design. I would say, "Bwaahahahahahaha", but that might come across as rude, which I don't mean to be. That reaction comes from my own painful experience. I thought the same thing when we installed a very pretty Kohler toilet with the Ingenium flush, which Kohler is very proud of. The toilet was such a nightmare, requiring constant plunging, that I finally ripped it out and replaced it with a Toto Drake that I bought for a fraction of the price of the Kohler, and which works fabulously on just a sip of water. It, too, shuts the flush valve (a simple flapper design) almost as soon as it opens. The effect is jarring at first -- you can't believe it's supposed to work like that if you are used to old toilets -- but the stuff really does go right down. There's no mistaking that that's the proper function of that toilet. Not so with the Kohler. It just plain didn't properly flush solid matter, no matter how much additional water you gave it on the flush. The bowl started with the right amount of water and the toilet was level, two other potential issues that you should check, but the bottom line was that the functional design of this very attractive toilet sucked, and Kohler has to have known that it did. They plainly just didn't care -- and they still sell this piece of junk.
I have to say that I was so happy with the decision to replace the toilet now having done it. It seemed like such a waste when I was contemplating it, which is why I took so long to do it. Now that it's done, I am so sorry I didn't do it sooner. Really, for the price of the professional help you will need to tweak one of your Kohlers to make it work as well as it can (which may not be so great in the end), you could buy and install a Drake yourself. And when you do and the thing works so amazingly-well, you will say, as I did, "Ahhhh. That's how the Kohler was supposed to function, but didn't." You will get, every time, that nice gurgle noise that you describe, and you will see that it can be done on 1.6 gallons and even on the 1.28 gallon EcoDrake, which often sells for basically the same price.
PS I worry about that Keeney valve. It says it "fits" 3" toilets and even says it "is" a 3" valve, but you correctly point out that it looks in the photo like it's got a much-smaller hole under the flapper, quite possibly 2". I wonder if an expert could comment on this but it would seem to me that that would degrade the power of the flush, based on the simplest fluid dynamics. If you're trying to tweak your toilet, wouldn't you want something like a 3" toto flush valve, assuming that the refill tube can be had in the right height? Or maybe the Fluidmaster universal 3" flush valve (2.8" throat diameter). Don't know why they wouldn't fit if the Keeney does, but, again, I would want to hear what one of the experts says about the concept and those products. Of course, because we are not convinced that the valve itself is the problem here, you might just be spending money to end up with the same poor level of functionality.