I am mystified as to why you are so certain of this point. The Rema 253 made no mention of any Tzorech at all.
I understand that you feel there is a contradiction between the Rema 253 and 276, and feel a need to make some distinction. However, your problem is based on a Hanacha that 253 is a Tzorech Gadol. A general rule in learning my Rebbeim taught me way back when is: when you have a problem based on a Hanacha, throw out the Hanacha!!!