May I request that you clarify this with HaRav Schachter?
Also you mentioned that the Rama makes no comment on the Mechaber's Pesak that matza ought not be 1 Tefach thick. Yet the Rama does say that matza ought to be made Rekikin in the PREVIOUS Seif. a little puzzling.
It seems that the Rama was only concerned about those making pictures on the Matza and therefore suggests that making it thin will delay the speed of Chimutz. However, in cases where there is no delay made in order to make pictures in the Matza, there is no insistence that the Matza be made as Rekikin, i.e. an Etzbah thick.
I spoke to Rav Schachter about this several times. He holds it is completely permissible and has nothing to do with mesorah. He only cautions that you need to have a good hashgacha on the matzah like you would on any matzah that you buy, and he is not personally familiar with any of the hashgachos on sefardic matzos. I personally eat the matzah that Rav Ben Chaim shlit"a eats - I trust his judgment on the hashgacha.