If the wife was sincere, wouldn't she have converted to Judaism?
Her children still wouldn't be Jewish, so there wouldn't be much point.
They can be raised culturally Jewish, but they won't be Jewish.
Nyet! If a woman converts by Orthodox standards to Judaism, her children are Jewish, since she is then Jewish by Orthodox standards.
If a woman converts by other standards of Judaism (Conservative, Reform), her children are only Jewish by the respective denominations, not by the Orthodox standards, which applies in Israel for purposes of a Jew getting Israeli citizenship, if he/she wants to emigrate.
But, being candid, I believe that when a Jew marries a non-Jew, much of the world considers any child "half Jew." What that means can differ, based on who is thinking it. The answer often, I believe, comes when that child chooses a mate in the future. If he/she marries a Jew, then the grandchildren are then Jewish by the world's thinking, I believe. If the child chooses a non-Jewish mate, then the grandchildren have joined "Gentile society."
Based on the level of anti-Semitism in a country historically, I believe, many Jews that married out saw that their children marry Jews, since the children were more accepted in the Jewish community, if they were raised in Jewish traditions. I suspect, that over two millenia, this resulted in many Jews with light eyes, blond hair, and other assorted non-Semitic features. Elvis was correct, I believe, when he said, "There's a whole lot of shakin' going on."