There is a dispute between the Mechaber who is stringent and the Rama who is lenient with regards to bishul akum and how much involvement of a Yisrael is necessary to wipe away the issur of bishul akum.
Chacham Ovadia in Yabia Omer (5:10:7) discusses that there are leniencies (limid zechus) for sefardim with a few tzeirufim of opinions such as cooking taking place in a Jew's house, an eved canani,shechirus, not oved avodah zarah.
From my understanding of the gemera and sugya the problem of bishul akum altogether is meshum chasnus (intermarriage). My question is if the one who is eating it doesn't even know who cooked the food then the problem of chasnus doesn't apply. The simple answer would seem to be lo plug (we don't differentiate), but we see that some Rishonim consider it not bishul akum in certain instances mentioned above. So why isnt the reason of stam food an issue. For example, if I ate by my friend's house who happened to have a non Jew cook food from start to finish shouldn't I be allowed to eat the food since I have no interaction with the non-Jew?