The problem with Gentile second wives is that under Jewish law those children would NOT be considered Jewish as Jewish identity is matrilineal. Now if the women converted, or the children converted at the appropriate age that would be different. In both Christian Europe and in the Muslim world, conversion to Judaism might mean a death sentence for the convert. Certainly the rabbi and other Jews involved in such a conversion would be in a world of trouble. I suspect, given what happened historically, that most of the "non-Jewish" DNA that found its way in to the Ashkenazic population came from rape of Jewish women by non-Jewish men. Of course, some non-Jewish women may have been "brought in" to the community but given the attitudes of the surrounding societies this was very limited.
For an example of this, look at the obvious "white" DNA in the US "black" population. For all practical purposes this was from black women being raped/used by white male masters, white women with black male partners was almost zero until recently. The reaction of Christians or Muslims to Jews defiling their women would have been pretty much what happen to blacks in the USA especially the south with this.
For an example of this, look at the obvious "white" DNA in the US "black" population. For all practical purposes this was from black women being raped/used by white male masters, white women with black male partners was almost zero until recently. The reaction of Christians or Muslims to Jews defiling their women would have been pretty much what happen to blacks in the USA especially the south with this.