Stephen is recorded by Luke (in Acts 7:8) saying :
... and he gave him the covenant of circumcision and thus he begat Isaac
[EGNT Literal interlinear]
Stephen's wording conveys that the significance of circumcision, in its inauguration, was closely linked to the begetting of Isaac. The wording is very emphatic : the one follows the other as a consequence, not just in time.
Paul in his epistle to the Galatians confirms this :
Abraham had two sons ... he of the bondwoman was born after the flesh ; but he of the freewoman by promise.
[Galatians 3:22, 23 - KJV minus the italicised words]
The significance of circumcision is the fact of a birth that was beyond nature (Abraham being ninety-nine years old and Sarah being past the age of natural child-bearing) and the begetting, therefore, was wholly as a result of the promise of God - with whom nothing is impossible.
Paul further confirms that circumcision is a figure of the new birth (a birth that is not of the flesh, the flesh is removed) :
Now we, brethren, as Isaac was, are the children of promise.
[Galatians 3:28 KJV]
In answer to your question (regarding the place of woman in this covenant) it is Eve who is, undeniably, the focus of the promise made by God when judging the Serpent :
I will put enmity between thee and the woman and between thy seed and her seed ; it shall bruise thy head and thou shalt bruise his heel
Which Paul plainly interprets in Galatians 3:16 :
He saith not, To seeds as of many; but as of one, And to thy seed, which is Christ.
Wherefore Adam called his wife 'Eve' - the mother of all living. That is to say the mother of all that would truly live, according to the promise. That is, of a life not of nature, not of flesh, but of promise and of Spirit.