Kohler One Piece Toilet plumbing

The most common problem that causes the symptoms you describe is a slow leak around the flapper. A very small amount of water drains into the bowl over time, lowers the tank level a slight amount and activates the float valve of a couple of seconds to refill the tank. What you should do before getting carried away dissecting the fill valve, is to check the flapper, clean it and it's seat well and be sure it is centered on the drain. If there are any signs of wear or cracks around the edges, replace it. Monitor the tank level between phantom fills, does it change at all? The second thing is to slightly lower the float so the water fills to apx 1/2 inch below the fill mark. This will eliminate the possibility that you may be losing water over the overflow, and tripping the fill valve.

These checks are simple, cheap and often solve a very common problem. Good Luck.

The overflow tube drains directly into the bowl, not the tank. The only way I can think that it would prevent the tank from filling is if it was cracked or had a hole in it and allowed tank water to drain into the bowl. If this was the case, you would notice the fill valve cycling if it was a slow leak or never shutting off if it were a large leak. The reason the tank fills quickly when you divert the bowl refill tube into the tank is that you are simply not splitting the water from the fill value between the tank and bowl any more.

It sounds like your fill valve assembly is the problem. The fill value has 2 outlets, one that directs into the tank, and the other that sends water into your bowl refill tube. My best guess is that something inside of the fill value got clogged with sediment, hard water deposits, or something else that restricted it. These are relatively cheap to replace and likely not worth the time to clean them out in my opinion.