A leak around the flapper CAN drain only part-way, because the leak stops when the water pressure above it is reduced. However, that's not normally what happens. Is possible, though.
A siphon resulting from the fill tube being stuck down the overflow riser is a very likely possibility. Give us a picture of the inside of the toilet. The hose should be clipped above the riser and the water flow pointed into the riser.
I don't know what you mean by flapper assembly. If you mean the whole flush valve assembly, and you removed the whole assembly and replaced it, then the problem isn't a cracked overflow tube. If you mean just the flapper and seat, but you left the overflow riser in place, then a crack in the overflow riser at about the point the water settles to is probable. Sometimes, the crack is vertical from the top to the place where the water settles. Often, it starts under where the clip for the fill tube attaches to the overflow riser. Take off the clip and pinch the overflow riser and see if you can't see it cracked and separated.
Good luck. Let us know how it goes.