No. The inherent stiffness (strain to stress ratio) of high tensile steels is the same as that of CrMo and MnMo. A set of frames brazed up out of those three materials with the available variations in wall thickness, will show load, shock, and torsionally-driven flexing commensurate with the differences in tube wall thickness and outer diameter. This assumes the joints are of equal rigidity in all frames.

What's different among the steels is strength. Better allows, such as Reynolds 853 or Tange Infinity, are stronger, not stiffer. This allows making tubes with thinner walls. Thinner walls make tubes more flexible, so to restore suitable frame stability (not TOO much flexing), the outer diameter is increased.