Here is my private opinion.
ASME Sec. III requires more severe condition for the materials with great safety than ASME Sec. VIII or ASME Sec. I. Basically Sec. III does not agree to use the quenched-tempered materials for low alloy steels but carbon steels due to the unstable structure at high temperature or on the weldments. Also Sec. III is greatly worried about Temper Embrittlement by Vanadium (e.g., V4C3 carbide) even though it can be highly reduced by Molybdenum.
In these reasons, Sec. III does not permit the use of SA508 Gr22 Cl3 (quenched and tempered) yet. However, in a near future, it is expected the limitation of Q-T treated or Vanadium-contained low steels will be relieved step by step in Sec.III and Sec.II part D.
Thomas