Whether homosexuality is fixed or changeable remains an open question. There are well-informed people on either side of the issue, but as of yet no incontrovertible proof of a “gay gene.” Whether the Jewish religious tradition is fixed or changeable, however, is not arguable — at least not for Torah-loyal Jews.
Skate past the first part. It's the last sentence which caused my head to tilt a little. Because I would also agree that "whether the Jewish religious tradition is fixed or changeable" is rather inarguable -- I just think the obvious answer is "changeable."
Unlike many Christian sects, Judaism has classically not engaged in Biblical fundamentalism. Rather, the Tanakh is only part of the official religious doctrine governing the Jewish community. Alongside it and carrying equal weight is the Talmud, which, in brief, is a corpus of interpretations and expansions upon Biblical doctrine extending for thousands of years.* The Talmud is basically comprised of countless Rabbis giving their own interpretations on what the Torah, the Tanakh, and other Talmudic stories mean, how they should be interpreted, extended, circumscribed, or modified. It is a vast menagerie of differing opinions, and together it paints a dramatically heterodox and pluralistic picture of what Jewish tradition "says". In essence, Jewish theology was the original common law method.
Through this, it is beyond obvious that Jewish law and tradition has changed, often dramatically, over the years, and is the furthest thing from "fixed". The Talmud is chock full of disagreements, with dissenting and concurring threads diverging into a host of different "schools". Indeed, for much of Jewish history there were two separate Talmuds (Jerusalem and Babylonian). The claim that Rabbi Shafran is making here is internally contradictory -- it itself is attempting to enact a dramatic change in how we think of Jewish traditions, replacing historic fluidity with modern (and dare I say, Christian-influenced) stasis.
* This also caused me to raise an eyebrow to the phrase "Torah-loyal Jews". That, to me, sounds like it should be the motto for Karaite Judaism, but of course, most Jews aren't and have never been Karaites. The dominant thread of Judaism is not "Torah-loyal", or at least, not exclusively so. What it is loyal to is the process of an evolving understanding and uncovering of divine principles as they instantiate themselves in an infinite number of social, historical, and political contexts.