I will start by briefly looking at the position of converts to Judaism (i.e. Jews by choice) in Jewish law and society.
Generally, Jewish society treats Jews by choice in an exemplary fashion. This is no doubt in considerable part due to the fact that it is considered unethical to remind someone that they were once not a Jew. The only time when Jews by choice are looked upon with suspicion is when there might be reason to doubt the sincerity of their motives for converting. The most common case I came across was when a non-Jew converts for the sake of marrying a Jew. However, this is not a particularly common problem (the suspicion not the marrying) and where a person converts for sincerely religious reasons s/he is usually greeted with great warmth and understanding.
The following case concerning a female Jew by choice illustrates the general attitude in Jewish law and values to "race". Jewish matrimonial law as it applies to cohenim (priests) contains a number of forbidden unions. One of these is that a cohen may not marry a non-virgin. Although this law applies equally to all Jewish females whether they are born Jews or not, in the case of female Jews by choice the prohibition is applied universally. However, this has nothing to do with their position as full Jews, but is a technical decision based on the difficulty of satisfying the burden of proof regarding their virginity. Considering that at the time when this law was promulgated it was common in some non-Jewish societies to involve one's children in a variety of sexual acts this approach seems perfectly reasonable. However, if at the time when a woman converts she has a daughter who is under the age of Jewish majority, the daughter can marry a cohen when she grows up. The reason is that the daughter is regarded in Jewish law as a born Jew. This is possible because Jewish ethnicity is based on kinship and not "race" or genetics. If Jewish jurisprudence gave any regard to "Jewish race" such a decision would be impossible.
What Jewish society has never done in its entire history is to segregate Jews by choice from born Jews in the way that Christian associations like the "International Messianic Jewish Alliance" (1) and the "Messianic Jewish Association of Great Britain" (2) does concerning so called "Messianic Jews" (Jews by "race") and their fellow "non-Jewish Christians" (non-Jews by "race").
This could explain why in my discussions with Jews by choice I found that it was mostly non-Jews and not Jews who rejected their position as full Jews.
In the case of Christian extremists there are further issues to consider. Firstly, it is highly likely that the sort of extremists who cast doubt on the position that Jews by choice are full Jews are as fascistic in their life generally as they are in matters of religion. Rightwing religious beliefs often go with a rightwing conservative outlook on other matters. They are thus likely to be genuine old fashioned racists.
Secondly, racism coincides rather too nicely with their theological needs for them to ignore its potential. If rightwing Christian extremists were forced to accept that Jews by choice are full Jews, where would that leave that darling of the right, the "Messianic Jew"? The simple equation is, no "Jewish race" = no "Messianic Jew". Without the "Messianic Jew" rightwing Christian extremists would be deprived of the joy of believing that "the Jews" are at last coming to "Christ". A term not of my making but which was coined by a liberal Christian friend to describe these Christian extremists is the "Christofascists".
Casting doubt on the idea that Jews by choice are full Jews no doubt also brings the Christofascist satisfaction in that s/he is often an incurable control freak. Presumably they are unable to cope with the thought that individuals have a right to choose their own religion or indeed to choose not to believe. Thus by denying that a Jew by choice is a full Jew they must feel that they are somehow back in control.