For the handful of maternal and paternal haplogroups in Ashkenazim that are closely linked to the North Ossetians and Turkic peoples like Bashkirs -- I mentioned 3 out of the 4 haplogroups of this nature in JewishGen message #667951 -- I would posit anew the migrational path I had proposed decades ago whereby a small number of Khazars who remained Jewish were living in what became Kievan Rus' and became a part of East Knaanic Jewry, intermarrying with assorted Romaniote Jews and Mizrahi Jews, and subsequently mixing with other Jewish groups including West Knaanic Jews and Ashkenazic Jews. The Khazars switched their language from Turkic to Slavic to Yiddish, and their writing system from Turkic runes to Cyrillic to Hebrew square script.
I cannot agree with your proposal that Turkic elements were incorporated into a Jewish community in southern Italy, such as a population of proto-Ashkenazim. Modern Sicilian Catholics, who sometimes have several segments of Sicilian Jewish DNA, don't show Turkic admixture in the calculators nor do they have Turkic uniparental haplogroups. There are no living people whose ancestry is 100 percent Sicilian Jewish so we can't test your hypothesis in the best way but I never saw a reason to believe they had relations with peoples from northern and eastern Asia, nor that any Khazar Jews or Avar Jews had moved into southern Italy, although small numbers of Khazar Jews did live at least temporarily in a more eastern area of the Byzantine Empire and in Spain.
As others have said, the Jews in 14th-century Erfurt were a very mixed group and some members had genetic affinities that others didn't, sometimes seemingly randomly, without a completely consistent pattern when it comes to non-Slavic elements like South Asian, North African, and even East Asian, other than for example the "EA_Devils_Gate" element that Michalis Moriopoulous incorporated into his custom Vahaduo model using G25 coordinates to determine the Erfurt Jews' admixtures.
4 of the Erfurt Jews from the Eastern Profile group scored that element whereas none of the Erfurt Jews from the Western Profile group did. "EA_Devils_Gate" is a Northeast Asian cluster similar to modern Ulchi of far-eastern Russia and other Tungusic-speaking ethnicities and to also forming a minority portion of Koreans and Japanese but not found in most Han Chinese, especially not the Southern Chinese where Ashkenazim got mtDNA haplogroup M33c from. So I think "EA_Devils_Gate" would be Turkic Khazarian in this case, and would be at the deepest root of the "Siberian" and "Mongolian/Manchurian" affinities for Ashkenazim in some calculators.
In Moriopoulous's experiment, some of the Erfurt Jews with "EA_Devils_Gate" don't score in the other East Asian element, "EA_Hanben", which appears to represent affinity to Hanben people from Taiwan, and visa versa. The EA_Hanben element is in 10 of the samples, this time including members of both the Eastern Profile and Western Profile groups as well as among what Moriopoulous calls the Mixed Western-Eastern Profile. In fact, the highest Hanben score (2 percent) is in a Western Profile sample (I13863).
Using Eurogenes K13's admixture estimates, Erfurt Jewish sample I13869 scores 3.08 percent Siberian and 2.06 percent East Asian, adding up to 5.14 percent. Her Siberian score is the highest of all the tested Erfurt Jews.
The only carrier of mtDNA haplogroup N9a3 (which probably originated with a Khazarian convert), sample I14740, scores 2.74 percent Siberian and 0.95 percent East Asian in Eurogenes K13. She has the third-highest Siberian score among these samples.
The authors of the Erfurt study cannot be correct in assuming that East Eurasian DNA came into Ashkenazim from a Slavic people. There's evidence in these samples as well as uniparental evidence that Slavic DNA in Ashkenazim is related to the Slavic peoples of Czechia, Poland, and perhaps eastern Germany (e.g., the Sorbs in the 14th-century Krakauer Berg DNA samples), rather than to those of Russia or Ukraine. Ethnic Czechs and Poles have very low East Eurasian DNA proportions so it can't explain what Ashkenazim have.
The Erfurt sample I14899 scores 0 percent Siberian and 0 percent East Asian but 21.37 percent Baltic in Eurogenes K13. This disparity would show what some of us already knew that the Slavic and East Eurasian elements in Ashkenazim don't come from the same ultimate non-Jewish source population if we went back to the 10th century. But the Slavic-admixed and Eastern-admixed Jewish populations would have started intermarrying a short time after that.
The Erfurt sample I13862 scores modestly in the East Eurasian elements but has 18.25 percent Baltic, again all per Eurogenes K13.