Christians did consider Judaism as a threat to Christian theology, but not quite in the way you think for disputations.

The problem Jews posed to Christianity was fundamentally that they were the religion around when Jesus was, who were promised the messiah of the Hebrew bible, yet rejected Jesus as that person. This is fundamentally the underlying tension that creates Christian antisemitism, and it leads easily towards positions where Jews are actively denying Jesus because they're terrible, or something like that.

Christians were very much a secondary audience in disputations. True, they were present at many of them, but strengthening their faith was a bonus. The primary goal was to proselytize to Jews. By achieving this goal, the conversion of Jews to Christianity would itself work as a faith-strengthening mechanism for Christians. Disputations were attempts to counter the theological problem Medieval Christians faced when encountering Jews, but they were not supposed to resolve that directly.

It is instructive to get into some of the details of what disputations were like. There are a few famous ones--Barcelona in 1263 is probably the most famous, which pitted Rabbi Moses son of Nachman (known in Jewish sources by the acronym-name "the Ramban") against a convert from Judaism to Christianity named Friar Paul. This one seems to have been won by the Ramban, though it was a bit of a pyrrhic victory for him as he was exiled. Friar Paul actually led the Christian side in another disputation later in Paris. Not all went so well for the Jews The disputation in Tortosa in the early 1400s, again between a convert to Christianity and a Rabbi (though not one so prominent as the Ramban) seems to have been followed with a flurry of conversions to Christianity, though the results of the disputation certainly worked in concert with an increase in anti-Jewish violence in the previous decades.

But, none of these were debates quite in the sense we'd think of them. Jewish attendance was generally compelled. This meant that if the Jewish side won, the status quo was maintained, but if the Christian side won, Jews would be persuaded to convert. You might wonder why Christians wouldn't just compel Jews to listen to priests preach at them, and it's a good question--they did, it's just less famous. Jewish leaders complained that this violated the norms of their residency in Christian countries, which largely fell on deaf ears, but a debate would pre-empt much of that. Jewish compulsory attendance at Christian sermons or disputations were closely tied with efforts to missionize among Jews more generally, and anti-Jewish decrees that would push them towards conversion.

To sum up, priests were not holding these disputations because they were afraid of Christians being persuaded by Jews. The point of them was to attempt to convert Jews, and in concert with other anti-Jewish policies, this was occasionally successful.

Sources:

• Stacey, Robert C. "The Conversion of Jews to Christianity in Thirteenth-Century England." Speculum 67, no. 2 (1992): 263-83.
• Bachrach, Bernard S. Early Medieval Jewish Policy in Western Europe. University of Minnesota Press, 1977.
• Chazan, Robert. Reassessing Jewish Life in Medieval Europe. Cambridge University Press, 2010.
• Chazan, Robert. The Jews of Medieval Western Christendom, 1000-1500. Cambridge University Press, 2010.