If it’s the married couple themselves who administer the Sacrament of Matrimony with priest as witness, does that mean that a couple can have others witness their vows without a priest and still be considered sacramental since they are the ones who administer this sacrament?
Generally speaking, it is not permitted and is invalid:
Can. 1108 §1. Only those marriages are valid which are contracted before the local ordinary, pastor, or a priest or deacon delegated by either of them, who assist, and before two witnesses…
There are some extreme circumstance exceptions:
Can. 1112 §1. Where there is a lack of priests and deacons, the diocesan bishop can delegate lay persons to assist at marriages, with the previous favorable vote of the conference of bishops and after he has obtained the permission of the Holy See.
Can. 1116 §1. If a person competent to assist according to the norm of law cannot be present or approached without grave inconvenience, those who intend to enter into a true marriage can contract it validly and licitly before witnesses only:
1/ in danger of death;
2/ outside the danger of death provided that it is prudently foreseen that the situation will continue for a month.
A bishop can only appoint a layperson to witness the vows if the bishops conference has voted to approve the use of such a measure *and *the Holy See has granted permission. Outside of that there needs to be either a danger of death without access to proper clergy or an ongoing extreme situation.
You are correct that it is the exchange of consent that makes a marriage, not the priest. After all, baptized Protestant marriages before a lay Justice of the Peace are valid sacramental marriages. The Church could permit anyone to be the officiant of a wedding, however She does not in order to preserve the integrity and importance of the sacrament.