R. Yitzchak Weiss (Resp. Minchat Yitzchak X, 23) questions the tentative nature of this “possible” leniency. He observes that the Shulchan Arukh’s language in quoting this rule indicates that one must make an effort to acquire a complete cup. In the understanding of the Mishnah Berurah (#10), this implies implies that failing to do so does not invalidate the process. If so, it would seem that leniency is not only possible but evident. He suggests, though, that R. Feinstein’s understanding, as his words indicate, is that a paper cup is even less appropriate than an incomplete one, and thus the leniency is questionable.
In further discussion, R. Weiss concurs with R. Feinstein, but for a different reason. The Rambam (Hil. Kelim 5:7) rules that a utensil that is used once and then disposed of is not subject to tumah. Apparently, such a utensil is not called a “kli”. This is an issue with ramifications for the laws of tevlilat kelim, as such utensils may not require tevilah [see Resp. Chelkat Ya’akov, II, 115; Resp. Minchat Asher [Goldberger] 5, 7, 8; Resp. Minchat Yitzchak, III, 76-78, IV, 114, V, 32; Resp. Iggerot Moshe]. Thus, this deficiency may also exclude paper cups for Kiddush. He concludes, though, that if no other cup is available, it is likely possible to decide to set aside the cup for continuing usage, and thus grant it “kli” status.