The vent won't cause those symptoms. It is more likely either something is trapped in the toilet, the wrong flapper is installed (so it stops before the design amount of water is dropped), the level in the tank is too low, or it's a lousy toilet.
What brand and model is it? Is the water level within about 1/2-1" from the overflow (or at the marked water height)? If you hold the flush lever to empty the whole tank, does it flush better?