In Passiontide, this prayer is to be omitted during the Mass and other public liturgical devotions, but what about in the Rosary and other private devotions? If it is omitted privately, is there some other Trinitarian prayer used instead? Or is this optional in private devotions and I'm overthinking it? The sources I've been able to find aren't clear about this. Thanks.
The Gloria Patri is only omitted liturgically. In private prayers, it is still said.
Just like we don't cover all our statues and pictures in our homes on Passion Sunday -- but we do in church. Because it's liturgical.
There is a big difference between prayers said with the mouth of the Church, and private devotions.
There is SOME overlap, for example: you wouldn't want to sing Easter hymns especially with "Alleluia" during Lent. But other than that, the Church just dictates how we will celebrate Lent, by dictating HER liturgy and changes which are to occur during Lent/Passiontide/the Triduum, etc.