I will say that it is documented in the discussions and various remarks by bishops that they didn’t like when secular motets were adapted into masses since the material upon which they were based was profane. So I wouldn’t adapt some secular motet into a religious piece just because the counterpoint is lovely. Reviving older religious works into the vernacular, though, appears to me to be something else entirely. Heath does this quite successfully and to all our great benefit.

The problem of different words simultaneously is in Motets (literally from 'mixed up'), when each of the voices sings a different text. For example, the Ite Missa est of the 'Messe de Tournai' is a polyphonic setting for three voices: One voice sings "Deo gratias", the second sings a French love song, and the third a Latin political text: and all three texts are underlaid and intended to be sung.

Looking at the Leonel Power score: It looks rather old, and not what one would consider a scholarly edition by today's standards. It is difficult to tell, especially without seeing the preface, whether the 'Alma redemptoris' text set in the Tenor/Cantus Firmus in italics in this edition is actually written in the MS or if it an editorial decision so that those reading could find the notes of the Cantus Firmus if they are unfamiliar with the chant. It is likely that the C.F. could be untexted in the MS. There are different theories of how such a work would be performed: Was the C.F. simply played instrumentally? Was the text set 'on the fly' by the singer, perhaps breaking the longer notes, or leaving out phrases? (The latter certainly can be seen in later polyphonic works where the C.F. only sings "Et in terra pax hominibus bonae voluntatis" in the amount of time its takes the other voices to get from "Et in terra" to "Gratias agimus tibi"; and then the tenor picks up again with "Domine Deus" along with the other voices--Such a scenario can be seen in Palestrina's Missa L'Homme Arme a5)

I would look for a better edition (which is hard to come by); or create your own, texting the Tenor as you see fit. This is a grey area, and different editors do different things.

I'm not sure that Trent is governing law here, even in a Tridentine Mass (for which the last word would be De musica sacra et sacra liturgia 1958), let alone the Pauline Mass (for which see the GIRM). For the TLM at least (all I will opine on), a vernacular text would be right out, esp. a secular one, but I don't think there's an active prohibition on multiple sacred texts. And if you have a scruple, then just underlay Mass text under that voice.

Post-Christopher Page performance practice is to sing untexted parts wordlessly. I prefer to have words. Ideally, you wouldn't break ligatures, but nobody will know or mind. Musicology serves the liturgy, not vice-versa.