I have never entered the EP debate on this forum, mostly because I cannot understand how those who claim to be confessional (British or Dutch - WCF 21.5 or Dort Article 69) could be anything other than EP or Inspired Only (or state an exception). But since someone else has brought that point up, I will make a few comments.
First, out of respect for the OP, which stated that this issue is similar to the head-coverings issue for him, my wife covers her head in worship because she fears God - maybe I Cor. 11 doesn't require it. But it might. Better safe than sorry. I sing only Psalms in public and private worship because I fear God. Maybe Paul is introducing the singing of uninspired songs in Colossians and Ephesians. But maybe he’s not. Again, I would rather be safe singing something I know God approves of (His Word) than something I am not absolutely (exclusively?) sure He approves of. The motif of our sacrifices to God having to be pure and strictly according to his revealed Word runs throughout all Scripture. In the previous dispensation it had to be a blood-flowing animal, not a plant (starting in Genesis 4 - or maybe 3.21 contra v.7?); it had to be the exact order set forth by God or it was considered strange fire (Leviticus 10 and elsewhere); it had to be a certain animal with no spot or blemish (pretty much all of Leviticus). Where is the evidence that God is now (in this current dispensation) willing to accept something that is less than pure as a sacrifice of praise? The fear of the Lord is not only the beginning of wisdom. The fear of the Lord is also the beginning of worship, as the WCF reflects in introducing Religious Worship and the Sabbath Day (Ch.21): “1. The light of nature showeth that there is a God, who hath lordship and sovereignty over all; is good, and doeth good unto all; and is therefore to be feared, loved, praised, called upon, trusted in, and served with all the heart, and with all the soul, and with all the might.” If you do not begin with the fear of God, your praise will not be unblemished. It is interesting that, by fearing God, I enter worship with no fear that my sacrifice of praise will not be acceptable.
Second, to one of the initial comments in this thread that the Psalter is insufficient and hymns are needed to speak explicitly about Christ - this, as has been noted, is a specious claim. I only add that, not only are the Psalms one of the Biblical texts most heavily laden with Christological content, and not only are they the most quoted by Christ Himself, but Christ’s name is literally all throughout the Psalter (Hebrew מָשִׁיחַ / mashiach = English the anointed one / Messiah / Christ = Greek Xριστός / Christos). Anointed One / Messiah / Christ is used throughout the Psalter - we just don’t translate/transliterate it consistently. But you can translate “anointed” in your Psalter to “Messiah” or “Christ” if it helps you (some Psalters and translations do - see Ps.2.2. in the current RPCNA psalter and in the Geneva). This would probably be preferable in Psalms that use this term and are directly quoted in the NT as referring to Christ (see Ps 18.50 = Rom 15.9; Ps. 45.7 = Hebrews 1.8-9; Ps. 89.20 = Acts 13.22).
Third, while I would not say I am strictly EP as I am comfortable with the Dutch and Calvin/Geneva’s singing of Scripture other than the Psalms (I believe the Colossians and Ephesians passages allow this by not saying just “Psalms” though I echo the thought that these passages are not both referring to public worship), I am more comfortable in an EP environment. As has been noted above, while there are inspired songs outside of the Psalter, when the latter was compiled by the Church in the previous dispensation, the Church included some (David’s song in II Samuel 22 = Ps. 18; Asaph song in I Chr. 16:7 = Ps 105) but not others (Moses’ song in Ps. 90, yes, Moses song in Deuteronomy 31-32, no). Solomon of course wrote over 1000 songs (I Kings 4:32) and the Song of Songs, but only 2 were included in the Psalter (72, 127). But there is no evidence that anything other than the Psalter (once it was compiled) was sung by our forefathers during communal worship in the previous dispensation. I am quite confident that Christ was singing from Psalm 118 and not the Song of Deborah on the night in which He was betrayed (the Talmud makes it clear that this was the tradition at the time, and this is not a modern opinion - you can find this idea in the opinions of numerous members of the Westminster Assembly, John Owen, Matthew Henry, Jonathan Edwards, David Brown, Alexander Hodge, to name a few).
Finally, to return to the original OP and the historical aspect, as someone who has spent most of their life in the US in the North, I have always felt (i.e. never had time to research) that the move away from the Psalter in the US was a direct result of the American Civil War. The South seemed to have the more theological conservative antebellum presbyteries (minus the obvious issue of chattel slavery - please don’t get this OP further off track with that topic) but then lost 1/2 million (out of less than 3 million) men in the conflict, including many who were devoutly presbyterian and reformed. I suggest that the vacuum left by these men’s deaths led to practical if not theological decline in the area of public worship (if anyone has resources/articles on this topic, I would be very glad to receive them) similar to what was happening in the North.