The Catholic Church is an Apostolic Church. It relies on the apostles, and their successors the bishopric, to transmit revealed truth through each generation until the end of time.
There is no way to prooftext the Bible and arrive at concrete conclusions because the Bible was never written for that purpose. The New Testament in particular is just the 4 Gospels, followed by an assortment of letters, followed by a prophetic text - Revelation - written in similar format to Zechariah. It is completely inadequate at addressing all of the minutiae of life. A living Church does that.
In this particular case, those verses are understood as a ‘discipline’. Meaning they are something the apostle wrote to his 1st century audience to be observed in the early churches, but it does not have to apply in all places and at all times.
How do we know this for a fact just from reading the Bible? We don’t. It doesn’t systematically flesh it out. Paul literally spends a few sentences of ink on it and then moves on. If you presented to him the notion that he needed to spell everything out legalistically and lengthily in his letters he would have found you eccentric. He was writing exhortations to different churches during his very dangerous and busy missionary life. A few of them - especially Romans - also contained some deep theology. Although we know that Paul, in addition to being a highly educated rabbi, was a mystic and experienced visions, it is doubtful he had a clue his words would become enshrined into sacred text and the liturgy for millennia. Christians were expecting Christ to come back any day. You have to rely on the apostolic authority which preceded Sacred Scripture, and which itself penned Sacred Scripture.
So in short: women don’t have to wear head coverings because the Church says so. Men can theoretically pray with their head covered. Why? The Church says so. We comply to her. As for those Catholics that don’t comply - or who consider the Church too many steps to the left or to the right for their personal liking - and who go about their own way: they will explain themselves to God.