During the more or less fifteen years I have following this website, it has over and over again been explicit that not everything we today call “cheese” is included in the gezerat of Gevinat haKum.
If you google, it is still possible to find the messages of the of the old board such as these following answers from the forum:
“Hard cheeses need a heshsher. Soft cheeses like the Mozzarella do not need a Hechsher, but you need to confirm all the ingredients are good”
“Can you please tell me if the Shop Rite Shredded Mozzarella Cheese (Part-Skim) is kosher for passover? R: Ok”
"The key to use is that any cheese that normally was made with Rennet and needs a form of artificial rennet or vegetable rennet to replace the animal rennet, this type cheese is what we are calling hard cheese, and that is where the restriction of Gevinat [haKum applies]… "
“Is Friendship Brand low-fat whipped cottage cheese kosher for Passover? It is OU, but does not say anything about Passover… reply: yes CYA.”
Mozzarella, as an example, was invented during the times of the rishonim and does not use rennett in any form (it is coagulated by a chemical process), therefore, it has no problem of Gevinat haKum. This has been the psak of Rav Abadi as brought by his sons.
However, your otherwise excellent summary of the laws of foods of a non-jew seems to imply otherwise.
Therefore, I would like to ask for a clarification about the matter and in particular:
1 - Whether what you wrote regarding cheese was intended to include cheeses like cottage and mozzarella and, therefore, cottage and mozzarella are forbidden?
2 - If yes, is this your understanding of the halacha or Rav Abadi’s and his son’s?
3 - If the latter, did they change position regarding what has been said for more than one decade? If they did, I would be really thankful if they can provide their reasoning.
4 - If the former, do you consider that Rav Abadi and his sons are wrong? Furthermore, what is your reasoning to include in the Gezera cheeses like mozzarella which do not need rennet and did not even exist in talmudic times?
Thank you very much and Shabat Shalom!