I didn’t mean your Chiluk of Be’etzem or external. I drew a parallel to the concept of Ischazek Issura. You are pointing out that they are not altogether the same, since by two pieces of meat you have Ischazek Issura and they are still Battul.
The main difference is that if you would be up close you would instantly know which is Muttar and which is Assur.
It seems like you were assuming in Tosafos that you have to be able to see the factor that makes it Assur or Muttar, which led you to wonder if that means Bris. What I’m saying is that in whatever way it might be, you would recognize the Jew from the others if you would be up close. That gives the Issur a steady existence and it is not swallowed up in the mix.
If I recall correctly, Reb Shimon Shkopp explains Parish and Kavua in a similar vein.