The offertory is equivalent to the Great Entrance of the Divine Liturgy, it is when the gifts are brought to the altar. The rubrics called for the priest to say silent prayers while the congregation sings a hymn. If the congregation is not singing, the priest may say the prayers aloud. Then the third option is a hymn using the words of the prayers of the priest, which is sung by the congregation and the priest. Since the words are already used, the priest need not recite the prayers silently.
Its not comparable to anything we do in the Divine Liturgy, the rubrics are just different.
I already looked this up a long time ago because I know we sing the hymn, “Blessed are you Lord” during offertory and once the priest, during a weekday, led the singing of that hymn during the offertory. It was the first time I saw that happen so I looked it up if it was proper for the priest to do so, and I found out that it is.