The priest is the minister of the Eucharist, true.
However, the priest is not always the minister of the sacrament of marriage. In the Latin Rite, those marrying minister the sacrament to each other, though the Church does require Catholics to be married with a priest or deacon officiating if possible. This is nearly always possible, unless you have been shipwrecked or are living in an area with no priests or deacons, etc.
Catholics can marry without priest in danger of death or if there will be no priest for a month.
The Bishop may designate a lay person to assist with marriages in areas where there are few priests.
I think things are different in the Eastern rites.
I recognize this is a subtle distinction that isn’t commonly known.
However, what Dan said earlier is correct. If even one of the couple Catholic, they must marry in the form the Church has established. When priests/deacons are available, that form includes priests/deacons. Because priests/deacons are general available, and few people get married on their deathbeds, most Catholics are married with priests or deacons assisting.