If the flapper is being held up by the chain, you are constantly dumping water, and be on the lookout for a surprise when you get your water/sewer bill, it could be huge.
Modern toilets usually do not empty themselves with each flush (the tank, the bowl should!) - they use that height to generate a more forceful flush. Think of a water tower on top of a hill - you get the pressure from the height.
On a pressure assist toilet, the water goes into a closed tank with a bladder in it and captive air. The water pressure pushes the water into the tank, compressing the captive air. When you flush, instead of gravity pushing the water out for the flush, it is the stored energy from the captive air (compressed by the water pressure) pushing it out. Thus, the water pressure from the supply is 'assisting' the water flow and the resulting flush.
If you have a typical flapper valve, it doesn't sound like it is a pressure assist toilet.
The static water height in the tank needs to be high enough to get the proper 'head' in a gravity flush; if it is too low, the water doesn't have as much pressure.