That sleeve in the photo is designed to keep the chain from bunching and kinking. It's not there to add weight. Some of the Toto flappers have and have had it, and others that allow the same amount of water through do not. The issue as far as how much water goes through is the buoyancy of the flapper, which is controlled by the amount and size of the side holes in it (as well as other design factors). On the adjustable ones, you adjust buoyancy by covering or uncovering holes and, in the particular tank with a particular-height water column, you thus get the desired amount of water through the valve. The same Toto flapper is used on some 1.6s and some 1.28s. For example, the TSU331s (Korky 2021BP) is used in the 1.6gpf Original Drake, as well as several "Eco"(1.28) toilets, including the Eco Soiree. In the Drake tank, it allows enough through the flush valve to result in a total consumption of 1.6gpf; in the Eco Soiree tank, with its water column and settings, it allows enough through in the flush to produce a total consumption of 1.28gpf (remember, part of the gallons-per-flush measurement is the refill water, so X.XXgpf is measured as the amount of water through the fill valve per cycle, not the amount let out by the flush valve). Other 1.6s and 1.28s have different flappers.
And of course in my post, I was talking about the bowl, not the tank. The Original Drake now uses a bowl optimized for 1.28gpf, regardless of whether you buy the 1.6 or the 1.28 version; the difference is the settings of the guts in the tank, or the guts themselves; the porcelain is the same.