Is Kyrie part of the Penitential Rite?

[Vico]

Well he also stated

It’s fascinating that the rubrics don’t clearly lay this out.

As he says, in the EF the Confiteor and the Kyrie are quite separated. In the OF they come one after the other.

So we have 4 situations where the Kyrie is ALWAYS omitted. 3 situations where it’s omitted if appropriate (whatever that means). 9 situations where the penitential rite is omitted (you forgot Weddings), but no note is made as to whether the Kyrie would be included.

The more I read, the more I feel that in those 9 situations the option to sing/say the Kyrie is still there and even in those other 3 situations. Whether in those 9 situations it’s an option or a requirement to include or maybe it’s not included and just not made clear, I don’t know.

When Is The Kyrie Appropriate?

It takes some judgement on the part of the priest. The Kyrie as used nowadays in the Roman Rite is penitential, so that would make it more appropriate for penitential seasons. “CIC Can. 1250 The penitential days and times in the universal Church are every Friday of the whole year and the season of Lent.” However from a historical perspective, and in the current eastern Catholic churches, as shown in the Byzantine Catholic Church, Particular Canon Law:
Canon 880 §2
§1. The special penitential seasons are:
1o. The Great Fast
2o. The Peter and Paul Fast
3o. The Dormition Fast [August 1-14]
4o. The Philip Fast [November 15-December 24]
Blessed Pope John Paul II said in Angelus (1999):
"Preparing for the Lord’s Birth this year means preparing to enter through the Holy Door, the symbol of the passage to new and eternal life, which Jesus Christ came to open to every human being.

This accentuates the penitential dimension, already present in the Advent season  and vividly recalled by the person of John the Baptist, who teaches, precisely,  that the way of the Lord is prepared by changing of one's mentality and life (cf.  Mt 3: 1-3)."

vatican.va/holy_father/john_paul_ii/angelus/1999/documents/hf_jp-ii_ang_28111999_en.html

So Advent may be considered penitential for the Latin Church, and therefore appropriate for the Kyrie.