David,
1) Just the fact that you have steel races and bearing elements all with the same CTE does not make the delta-T issue irreverent. The sizes of each element is different, so the expansion of each element is different. At least that was the primary issue back when I was building and designing (in the evolution of my career) rotary elements in hot gas-flow environments.
2) Think about the heat treating process. First there is hardening where you heat the metal up to some soak temperature, increase the temperature to a second soak temperature (usually just above the Curie Point), and then quench it to force the crystal structure of the metal to remain in the (high temperature modified) form. This is usually followed in modern commercial practice by a quench & soak at some seriously sub-zero (cryogenic) operation that, as I was taught in the 70's, converts more of the crystal structure into the high temperature modified form before being returned to room temperature. We now have a part with high hardness, high (stress-based) strength, and high brittleness because of the internal strains (i.e. stresses) inherent in the unbalanced high temperature modified crystal structure. Those stresses (strains) are relieved by tempering. The higher the soak temperature used in tempering, the more of the stresses are relieved and the hardness, strength, and brittleness are reduced. (And, in modern practice, tempering often includes more cryogenic processing.) The variations in temperatures and time of soak vary by the chemical composition of the alloy.
High-speed steels are a class of alloys that retain their hardness, strength, and reduced brittleness (toughness) to higher operating temperatures than traditional carbon steels. Other classes of alloys (hot-work, tungsten alloy, etc.) have been developed for similar reasons. Thus, if you need a part to retain (say) Rc-62/60 hardness at some elevated temperature, it is more a matter of selecting the appropriate alloy (with the concomitant heat treating regimen) than applying a process to an alloy without the ability to maintain physical properties at the operating temperature. It would not surprise me (not being an expert in this field) that there is some secondary thermal processing required to establish properties at a given elevated operating temperature, but I have never heard of such a thing being called stabilization.
At the risk of sounding like a hopeless pedant, I looked up heat stabilization in my 1984 Edition of the (26 volume) ASM Metals Handbook and found nothing appropriate to bearings. (Most of what I found deals with alloy composition for stability at various operating temperatures.) I have the standard machinist/tool & die/blacksmith background in metallurgy of my apprenticeship coupled with 2+ years of having a formally trained and well-experienced METALLURGIST (who led metallurgical R&D for the US Army for more than two decades) as my adviser in college. I have held my own with metallurgists over the years in a wide array of projects -- but I am not a metallurgist -- only a general-purpose mechanical design engineer.