The first hymn of the Wednesday Vespers in the Syriac tradition addresses the Mother of God directly ("to her"), asking her to pray "with us" and "for us" to her Son. The understanding of "prayer" here is not worship, but more like "communication", "asking", etc. Furthermore, we believe that it is Christ himself who is the sanctifier of the saints (e.g., festal Trisagion, Vespers, tone
, and who grants not only the answer to their prayers, but also the efficacy of their intercession before him. Many of the hymns for the saints in the canonical offices, in referring to their relics, affirm that "(the power of) Christ dwells in their bones". While I can't speak for the Coptic tradition, I'm pretty sure it is the same concept for them and all the rest of our Churches.
I, too, believe that the statements of Fr Anthony (disclaimer: I didn't listen before posting) reflect the conflation of "prayer" with "worship" or "adoration". As someone alluded above, all worship is prayer, but not all prayer is worship. Many equate the two, leading them to think our veneration of the saints is simply a thinly veiled polytheism. Sadly, this is practically (if not intentionally) how some of our faithful practice their faith. I've heard stories from priests of parishioners who'll say something like "I don't have much faith in Jesus, but I have great faith in Saint Insertnameios: he always answers my prayers!" (prompting at least one priest to respond "When you say such things and mean them, you turn the saint's heaven into hell"). Perhaps this tendency among some is a reason why you'll occasionally find Orthodox speakers "downplaying" the "cult" of the saints. But in doing so, we risk diluting the authentic teaching of our Church on salvation, sanctification, theosis, etc. We can affirm that "it's all about Jesus", placing the veneration of the saints in its proper relation to Christ, without belittling the economy of salvation by which Christ sanctifies those who believe in him, re-creating them through incorporation into his body and cooperation with divine grace.
Unfortunately, and in saying this I mean no slight at all to Fr Anthony, it seems that a lot of our recent teaching and preaching is "reactionary". Instead of going on the offense and taking the time and effort to teach the truth, we're too busy being on the defense, cleaning up the mess caused by heterodox presuppositions. When we do the latter, it's tempting to take the easy way out and simplify things. Simple is good, but sometimes it'll just create a different but equally annoying mess. Truth is better.