jungerman1- The “hetter’ of cooking from jom tov to shabbos is based on “ho-il” (pesachim 46B). This means that you can cook on jom tov because you may have “orchim” -guests coming unexpectedly and they want food. so you cook in the antcipation of their arrival. (BTW- this is why many rishonim will say that you cannot cook this just before night because there will be no time for guests to arrive) The idea of eiruv tavshilim is that you started cooking before yom tov and hence you are only concluding your cooking on jom tov. basically ,it is a “hekker”. see Shulchan aruch 527).
toi and Sam2- I like your answers because you may get guests woh eat “gebroks” and they have to have food too.