Pawlos, the Ge'ez form of Paul, begins with this letter, and Petros, the form of Peter, begins with it in the fifth order. The two most frequently named apostles of the New Testament both have names that a Ge'ez writer could not spell with an inherited letter, because the language had turned every Semitic p into f before those names arrived.
A language in that position has three options. It can substitute the nearest sound it has, which is what happened to the Greek pascha when it became the Ethiopian Fasika, spelled with af. It can leave the word out. Or it can add a letter. Ge'ez did the first for some words and the third for others, and the result is a small inconsistency in the lexicon that is a fossil of the moment the Greek text arrived.
Payt therefore belongs to the same layer of the language as wengel and the rest of the Greek Christian vocabulary. It is a letter that exists because a body of foreign names had to be written down accurately in a script that had no sign for the sound in them, and the fact that it was added rather than avoided says something about how carefully the names were being handled.
The lectionary is the reason it mattered. Readings from the epistles are appointed through the year, and a name standing in the rubric of a service every week has to have a settled spelling. A letter added for a handful of proper names is a letter added because those names are read aloud constantly, in front of a congregation, by someone who cannot improvise.