@ 260 lbs + most any tire will be 'supple'
and paper thin sidewalls would become a negative
Suppleness has nothing to do with payload, it's an inherent property of the tire itself. Yes, it's relatively less important with heavier riders, but ANY increase in sidewall stiffness will increase rolling drag.
As for paper thin sidewalls, again not a rider weight related consideration. The structure and pressure rating of the tire is purely a function of the body plies. Rubber is added to the sidewalls purely to protect those body plies from weather and road hazards. Riders riding on paved roads at proper inflation don't need to worry about the sidewalls. However, if you have gum (para) covered sidewalls, be aware that these are subject to UV or weather damage, so periodic inspection may be called for.
IMO- and biased by 50+ years riding tubulars, the best --- most efficient --- tires for road use are the treadless skinside tires offered by the major brands.