Except that, in the literal translations the exhortation associated with 'shall' is that the man must pay the bride price. The marriage already exists; the expectation is that the man will then formally acknowledge the marriage and make restitution for having attempted to steal something from the woman and her father. It's not about whether or not they get a license. If copulation requires payment of a bride price, then ipso facto she is already a bride.Now, I could be wrong when it comes to the Hebrew. There could be a translation issue. But in english it is abundantly clear that the requirement is that the man will in the future or immediately afterwards endow her to be his wife.
In some senses, the desire to add human tradition in the equation beyond the scriptural assertion that the one-flesh relationship constitutes a marriage in the eyes of Yah reminds me of the degree to which many Christians will twist themselves into pretzels trying to take credit for their own salvation, placing the onus of importance of how salvation occurs at the moment of formal acceptance. This elevates faith to be a work that gives the primary credit to the individual, whereas the credit for salvation belongs solely with Yah and Yeshua, as their Gift occurred in the past, and even the ability to have faith only follows Yah having written it previously on the individual's heart.
Man takes virgin; they're married. The Exodus and Deuteronomy passages are only concerned with the man taking full responsibility for having made them one flesh. Remember, too, that 'endow' and 'endowment' fully imply something related to payment.