Please notice the tenses here in what you quoted: "... as if He who begot were not one, He who was begotten, another, and He who proceeded from both, yet another."
This is not, as is the filioque heresy, a discussion of the eternal hypostatic procession of the Holy Spirit. It is instead a reference, at least apparently, to there being three distinct persons and to the Son's mission in time. For example, Holy Scripture tells us, "And when he had said this, he breathed on [them], and saith unto them, Receive ye the Holy Ghost" (Jhn 20:22). That, it appears to me, is what St. Leo is referring to. The Creed, by contrast, refers to the hypostatic eternal generation of the Son and to the hypostatic eternal procesion of the Holy Spirit, an entirely different subject. Of this, Jesus Christ Himself said, "But when the Comforter is come, whom I will send unto you from the Father, [even] the Spirit of truth, which proceedeth from the Father (Jhn 15:26 )".
This seems to be an interpretation very hard to maintain, in light of the fact that St. Leo is explicitly explaining why the Persons are differentiated, a matter of their eternal relation. Moreover, how would a temporal procession of the Spirit be congruous with the eternal begetting of the Son by the Father? "who begot .. He who was begotten ... He who proceeded".