Cheese---Special Issur?

I read in the Kitzur that cheese of a non-Jew is forbidden even if the Jew supervises (which would be sufficient for milk), unless it is owned by the Jew as well at the time of its making. Is this the generally the accepted P’sak or do we hold differently? According to this opinion is there a special Issur for cheese, or is there simply more of a reason to be concerned about cheese production (i.e. animal rennet)?

The reason our sages prohibited milk of a non-Jew, is because we fear the gentile may have mixed in it milk from an impure animal that is a prohibition from the Torah. In a case that a Jew sees the milking, the milk is permitted. On the other hand, cheeses of a non-Jew are prohibited because they coagulate (Ma’amid) it in rennet from animals which they slaughter and are Neveilah. (There are additional reasons for this prohibition, see Tractate Avodah Zarah 35a). The Shulchan Aruch and Rema rule that the custom is to prohibit all of their cheeses even if it was not coagulated in rennet at all. The Rema adds that if the Jew sees the manufacturing of the cheese and the milking, it is permitted, and thus is the ordinary custom in all these states. The Shach, the Gra and the Pri Chadash, disagree with the Rema. According to their opinion, the cheeses are prohibited even if the Jew sees everything, as long as the Gentile is the one making the cheese. The proof is that even if the cheese was coagulated with grass it is still prohibited because of Lo Plug (standard rule without exceptions). According to them the cheeses will be permitted only if the Jew himself will coagulate the cheese or if the cheeses will be under his ownership. Based on these Poskim the Kitzur Shulchan Aruch ruled that the cheeses are permitted only if owned by the Jew at the time of its making. Nevertheless most Poskim rule like the Rema, that if the Jew sees the manufacturing of the cheese and the milking, it is permitted, and it is not necessary that it should be under his ownership or that he should be the one who makes the cheese. According to their opinion the prohibition of cheeses applies only when there is no Jew overseeing the gentile. [קיצור שולחן ערוך סימן לח סעיף יד, ע"ז דף לה. שו"ע יו"ד סימן קטו סעיפים א,ב, וברמ"א, ש"ך ס"ק יט, כ, באור הגר"א ס"ק יד, גליון מהרש"א ובית מאיר שם, פתחי תשובה ס"ק ו, חכמ"א כלל סז סעיף ז, ערוה"ש סעיף יט, כף החיים ס"ק לב, שו"ת אגרות משה יו"ד א סימן נ, ועיין בספר הכשרות של הרב פוקס פרק ח].