Oğur vs. Siberian?

I was listening to this Tuvan song (and now you’re going to listen to it too):

And I took special notice of “doştuğ” being equated with “buzlu”. Certainly cognates of Istanbul “buz” are in use in Siberia, as even the faraway Sakhas have “буус” (buus). But “doş” doesn’t appear to be a Mongol word, as the Mongol for “ice” is “мөс” (mös), which if anything is closer to “buz” (and its common Central Asian variant, “muz”) than to “doş”!

The closest word I could find (other than the same lonely word in neighbouring Siberian varities) was “dolu”, meaning “hail”, which surfaces as “dolu” in several Turkic varieties, including Tuvan itself.

One is also tempted to link “doş” to “daş” (stone), so that this might be some rounded varient that somehow was spread in its Oğur (or Lir Turkic) form among the vast majority of Turks (as hail) while it was preserved in its authentic Common Turkic (or Shaz) form in some Siberian varieties only. But this requires several enormous leaps of faith:

  1. We have to assume that there was some vowel rounding of “stone” to achieve the meaning of “hail” back before the split between Oğur and Common Turkic was complete.
  2. Even if we assume dolu/doş as an independent root, we have to assume that the Oğur term for “hail” gained currency and predominated among most Turks except some Siberians, who shifted the meaning of the native stone-based word for “hail” to “ice” while accepting its Oğur equivelant as the term for hail.
  3. THEN the only remaining Oğur speakers, the Chuvash, themselves lost any trace of this theoretical proto-Turkic root (or theoretical Old Turkic mutation of “stone” which meant “hail”).

But if not this, then what?