> The only part of the battery tech that could affect range is the weight.

Weight is a pretty low factor for cars, sub-percent (aging wheels did a comparison using a pickup empty versus loaded with a pallet of shingles, though with a more efficient vehicle the influence of weight probably shows up more).

Energy density (amount of energy per unit of volume) is a much bigger factor than energy specificity (amount of energy per unit of mass), it means you can either cram more energy in the same volume for more range, or have a lower vehicle with better aero.