Most wine is mevushal. Plus, if a person is a tinok shenishba, would they have the din of a mechallel shabbos b’farhesya? And, Reb Moshe paskens that mikar hadin, a person that is Mechallel Shabbos that touches wine does not prohibit the wine mikar hadin.
Also, even if they cooked swordfish or sturgeon, that wouldn’t treif up the pots and pans, since there are shitos which say these fish are kosher (R’ Hershel Schachter, the Noda bi Yehuda, Knesset haGedolah, and the “mesorah” experts Greenspan and Zivotofsky, for starters).
My feeling is that objectively, there would be little to cause their kitchen to be treif.