I have an American Standard Vent-Away series 2003 in my master bathroom (house built in 1976). Most of the incoming water seems to go right into the bowl, not the tank. The flapper seems to seal fine. In the bowl, the water just swirls and drains slowly.
I am willing to pay the outrageous price for the parts ($200 for the fill valve alone!) because the toilet is dark brown, which matches the bidet, tub, towel hangers, and trim of this Tudor-style house.
Although you state that the flapper seals fine, have you checked the seat which the flapper seals to? These can/and will often get small nicks in them which will let water seep by with the flapper seated on the seal. Turn off the water to the toilet, allow the tank to drain as much as it will, lift the flapper, and run your finger over the top of the seat--lightly. I have had nicks in the seat cut my finger while checking. Another option would be to hold the flapper down with moderate pressure while someone opens the toilet supply valve. See if this will allow the water to run into the bowl or not. If it does not run into the bowl, then the flapper is not seating properly. If it does run into the bowl, I'll be waiting for the answer to the problem as you are. Good Luck, David