Maybe I'm completely wrong about this. All I can say is that there's no actual evidence the 3-part Kyrie was sung alternatim.
We can't argue from (continental) Italian or (obsolete) Sarum-rite practice, since those were both basically foreign to what was going on in English recusant circles.
Is he advising (chant small letters, polyphony big) something like
KkK
cCc
KkK?
I might call it a touch of excessive rubricism/legalism to ignore what Byrd put down on paper in favor of some idea of greater liturgical propriety - but (big disclaimer) I'm just a musicologist, not a liturgist, and I tend to take my 16th-century brethren at their words.
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