Marv,
I noticed the ASTM specifications in your post and looked these up in my ASTM standards. The A106 and 513 standards are not listed under structural-steels. These are listed in two different ASTM volumes (01.04 for structural steel and 01.01 for steel pipe, tube and fittings). The above standards are not classified as structural by ASTM regardless of the yeild/tensile values.
Dimensional properties may vary as well (i.e: the radii of corners on square/rectangle shapes) compared to those of non-structurally approved materials. The elasitc/plastic behavior, brittle fracture concerns due to manufacturing processes, all of these may affect the performance of the material. I do know many materials with the same tensile/yeild propeties do not perform identically.
I also had a client who used some "non-standard" structural materials (and weld configurations) and I was constantly qualifying welders and procedures to AWS B2.1-XX. I didn't mind as it was always a learning experience for myself and others. This specification was very handy for these types of welds and materials.
With regards to your original post, I don't know why more materials are not specified. Availability and cost may be other factors as to what is prequalified and what is not.
ASTM A512 MT1010, ASTM A513 MT1020, ASTM A519 Gr 1018 and ASTM A120
These are the specifications that you had posted in your February 7th post. I found no reference to A106 and as such my comments were not directed to that specification. As for A513 the only time you have tensile strength and hardness specified is when you specify supplemental requirement S5 in your purchase order and MT1020 shows an ultimate tensile strength of 52 ksi with an elongation of 12% and a yield strength of 38. Of course I am looking at ASTM A513-97a which might not be the latest revision.