Rabbi Yonathan arrives at the same conclusion as Rabbi Yoshiyah — that a non-Jew may perform labor for a Jew on the festival — but takes a completely different route to get there. His method is the kal va-chomer, the argument from lesser to greater.
He starts with Shabbat, the weekly Sabbath. The Torah states in (Exodus 20:8), "Remember the Sabbath day to keep it holy." Shabbat is the more severe day — its violations carry harsher penalties than festival violations, and its restrictions are more comprehensive. Yet even on Shabbat, the Torah does not explicitly prohibit a non-Jew from doing work on a Jew's behalf.
If the greater day — Shabbat — does not forbid a non-Jew's labor for you, then certainly the lesser day — a festival — cannot forbid it either. The logic is airtight. Whatever Shabbat permits, the festival must also permit. Rabbi Yonathan does not need to compare verses or parse passive constructions. The hierarchy of sacred days does the work for him.
So what, then, does the phrase "all labor shall not be done in them" actually accomplish? If it is not prohibiting non-Jewish labor, what is it prohibiting? Rabbi Yonathan reads it as reinforcing the ban on a fellow Jew — your neighbor — performing labor for you on the festival. The verse targets Jewish mutual assistance, not the actions of non-Jews.
Two rabbis, one conclusion, two entirely different methods. This is the richness of Talmudic argumentation.