The Son is not a separate principle from the Father. I already quoted the Fourth Lateran Council which specified that it is the Father alone who generates.
Rather illuminating for me in the process of checking my work on this topic has been a
tract prepared by the Pontifical Council for Promoting Christian Unity at the request of Pope John Paul II. One thing it notes is that the Western Church did not actually have the Nicene-Constantinopolitan Creed until around the time of the Council of Chalcedon, and their Trinitarian theology therefore developed independently, and most importantly, in Latin.
The Latin word
processio does not refer to taking origin as the Greek word
ekporeusiV (I copied this from another source, so apologies if it's misspelt) does, but rather it refers to consubstantiality, which, indeed, the Father and the Son have in common with the Holy Spirit, and the Father in common with the Son, and so forth.
When the NCC was translated into Latin,
ekporeusiV was translated with
processio, thus unintentionally conflating the two ideas. To the Latins, it was wholly necessary to emphasize the consubstantiality of Father, Son, and Holy Spirit, precisely to avoid the subordination of the Holy Spirit. Ironically, when the Byzantine Church heard the translated NCC, they heard a clear subordination, as illustrated in our dialogues even today. Oops!
Perhaps a good takeaway is that East and West developed their theologies in response to different problems, and consequently have different trains of thought. For the West, the importance is not double-procession or subordination, but consubstantiality. For the East, the importance is the Father as origin of both other Persons.
Which brings me to...
Yeah, a full-out debate about the Filioque may be more suited for the St Justin Martyr Hall. I'm mostly here to explain and defend the Catholic understanding of the Filioque, because I saw a lot of Orthodox folk on here, but not many Catholic folk.
I cut the rest of your comment for space, because that stuff is also more suited for the St Justin Martyr hall, or perhaps for PM. I would like to clarify Thomas's position on whether the Son alone became Incarnate: yes, he says later, the Word (which he uses as a synonym for the Son) alone became Incarnate, and in fact that the Word became Incarnate is most fitting. You can read his full thought process on it in the Summa, Third Part, Question 3, Article 8.
Ultimately, as I think I mentioned before, I think adding the phrase to the Creed was imprudent, but not necessarily heretical in light of linguistic and theological differences between East and West (even before the schism). I think it'd be best to remove it provisionally, especially if such will effect a better union between our Churches, then look at it with fresh eyes from both East and West.
But, what do I know.