Your argument is circular on 1 Tim. 2, and is not supported by Scripture.
Adam and Eve were given as to the reason for Paul's previous comments, Because Eve was the first in the transgression. If she had been properly in subjection to her husband, she would have called upon him before making any final decision; instead, desiring to do good for her husband, she became part of the problem.
While Adam and Eve were indeed the first husband and wife, their placement by Paul, in this Epistle, goes to the subject of Eve. Abraham and Sarah, are being used in a positive example, to which husbands and wives should follow. They are not being used in a negative example.
It is probable that the women in Ephesus were being deceived by some of the false teachers and this had a local application. They should submit to the judgment of their husbands who were trying to correct them and deliver them from the false teaching they were falling into. Paul might have been instructing them Timothy to tell their husbands to correct them and the women should be submissive to that correction. Maybe their false teaching included dressing like a harlot for attention.
There is a very real possibility that this is a local application addressing a local problem not a ban on all women being used in preaching and ministry offices in the church for all time which should be obvious to a Spirit Filled saint that would not be the intention of Paul's words here. But hundreds of years of oppressive false teaching using these verses is being broken by becoming more biblically literate.
Genesis 2:18–25 Some expositors have taught that all women should be subordinate to adult men because Eve was created after Adam to be his “helper” (NIV; “help meet”, KJV). Yet the word ezer (“helper”) is never used in the Hebrew Bible with a subordinate meaning. Seventeen out of the twenty times it is used, it refers to God as the helper. Eve was created to be a help (kenegdo) “suitable” or “corresponding to” Adam, not a subordinate.
Some argue that God created men and women with different characteristics and desires, and that these differences explain why leadership roles should be withheld from women. Others attribute these perceived differences to culture and social expectations imposed on children from birth to adulthood. Physical differences and distinctive biological functions are obvious; but it is only by implication that gender differences can be made to suggest leadership limitations.
First Timothy 2:11–15 The meaning and application of Paul's statement, “I do not permit a woman to teach or to have authority over a man; she must be silent” (1 Timothy 2:12), have puzzled interpreters and resulted in a variety of positions on the role of women in ministry and spiritual leadership.
From the above survey of passages on exemplary women in ministry, it is clear that Paul recognized the ministry of women. There were obvious problems in Ephesus, some relating to women. Some women were evidently given to immodest apparel and adornment (1 Timothy 2:9). The younger widows were “into the habit of being idle... And not only do they become idlers, but also gossips and busybodies, saying things they ought not to” (1 Timothy 5:13). In his second letter to Timothy, Paul warned against depraved persons (possibly including women) who manipulated “weak-willed,” or “gullible,” women (2 Timothy 3:6).
A reading of the entire passage of 1 Timothy 2:9–15 strongly suggests that Paul was giving Timothy advice about dealing with some heretical teachings and practices specifically involving women in the church at Ephesus. The heresy may have been so serious that he had to say about the Ephesian women, “I am not allowing women to teach or have authority over a man.” Other passages show that such exclusion was not normative in Paul’s ministry.