A better explanation, given that M124 is less rare among Mizrahi than Ashkenazi, and that Mizrahi inhabited areas with heightened M124, is that the Mizrahi gained the gene, after splitting from the Ashkenazi, via intermarriage in West Asia. This explanation is supported by the fact that M124 is considerably more common among the (non-Ashkenazi) general Jewish population than among the priesthood (“Cohanim”) – the latter being less likely to intermarry:
The distribution of haplogroup frequencies for all haplogroups present in Ashkenazi and non-Ashkenazi Israelites (top) and Cohanim (bottom) at a frequency >5%. The following haplogroups are not shown: C-M216, E-M96, E-P2, E-M81, F-P14, I-M170, I-P37.2, I-M223, I-M253, J-M304, L-M20, N-M231, Q-M242, R-M173 [1]
Given that we have no evidence that M124 “dominantly enter[ed] the Ashkenazi pool” in the first place, this would seem to be an absurd question.
Do you have any evidence for this hypothesis?
Addendum:
They are only “ancient” to the extent that they both date back to the Jewish diaspora, and Ashkenazi and Sephardi are not the only groups – Mizrahi Jews are a separate group:
It is likely in fact that Mizrahi, as they would likely include some descendants of the Assyrian exile and the Babylonian captivity, are the older population (giving more time for local admixture).