For many Jews that fleshly cut affects all it needs to affect just by being performed: it is sign and signified unified in the flesh of the Jew so cut.
Not sure what you meant by "sign and signified unified in the flesh of the Jew"
In Genesis 17, God says that all Abraham's spiritual offspring, not, mind you, his physical progeny, will be circumcised. It then implies his physical progeny, starting with Isaac, will be inscribed with the "sign" of the covenant. Being inscribed with the sign of the covenant is said to "guard" שמר the covenant until the spiritual seed arrives.
Exegeting the earlier part of the chapter, where Abram becomes Abraham, Rabbi Hirsch, Rashi, and many other sages, point out that Isaac was already promised to Abram in Genesis 15, such that the name change implies a different covenant, or a different element of the covenant, than what was already promised to "Abram" in Genesis chapter 15.
Isaac was already promised to Abram prior to the new covenant established in Genesis chapter 17. The Jewish sages (notably Rabbi Hirsch, Rashi, and Abarbanel) are aware that Isaac is not the primary focus of the covenant established in Genesis chapter 17. Isaac is Abraham's natural progeny. The covenant of circumcision implies a new, spiritual progeny.
Abraham, Isaac, and Jacob, and Israel after them, are the guardians of the sign of the covenant (its mark is in their flesh). They are not necessarily the primary target of the covenant. They're supposed to "guard" שמר the sign, the emblem, until someone, or ones, arrive to show what the sign, guarded by Israel for all those centuries, signifies.
John