By this logic, OP, one could then say that the spirit "proceeds" from others in the "mediatorial" manner when you consider Matthew 10:20 "for it will not be you speaking, but the Spirit of your Father speaking through you."
Orthodoxy tends to stay with the basics. The Divine Economy is eternal, thus the Word is eternally begotten by the Wisdom, and the Spirit eternally proceeds (wells) from the Wisdom. That never changes, always keeping in mind that God is transcendental.
Instead, you are thinking of the Pentecost -- a singluar point in time. Open up a little more; you are getting there. :-)
I agree with your point; good citation on Matthew 10:20.
However, even granting a "mediatorial" procession of the Spirit through the Agency of the Saints, you still have to admit the Son as the "First Mediator" of the Procession of the Spirit at the very least -- A.) Temporally, because (being Eternal Himself) He obviously is the First Sender; "Before Abraham was, I am"; and B.) Causationally, because unless the Son first send the Spirit to the Saints, the Saints have not the Spirit to send out into the World.
As such, I still think it is Biblically-correct to say that the Spirit proceeds "from the Father, through the Son". I think that such a formula does the best possible justice to John 15:26, even admitting the secondary-mediation of the Saints described in Matthew 10:20.
However, I admire the Eastern Orthodox desire to stick with the Scriptural basics as regards the economy of the Trinity -- which is why I am uncomfortable with the Roman-Latin filioque since, its meaning simply being "and the Son", it does not in and of itself explicitly and carefully preserve the apparent economic distinction of Trinitarian Offices evidenced in John 15:26.