I find it interesting to think the original accompaniment to the Psalms was David’s stringed instrument, the forerunner of the modern day guitar.
The organ is said to have originated in Chaldea or Greece, where it first appeared as Pan Pipes or Syrinx and was used for public celebrations. It was not adopted for use in the church until A.D. 450. It did not appear in Rome until the 7th century (where it was used to improve the singing of the congregation.)
The organ began life on a secular note and has now replaced the stringed instrument engaged by David in early liturgical worship.
You might have missed the post (post 151, page 11) providing the actual reason/history for the use of the organ during liturgy which Benedict XVI gave in an article/lecture to the Church Music Department of the State Conservatory of Music in Stuttgardt, Germany. Pnewton also provided a link to the entire lecture. It’s well worth reading thoroughly - so much information to drink in.

I really enjoyed it myself.
musicasacra.com/theological-problems/
Anway, the organ wasn’t employed to improve the singing of the congregation. In short, it represented “all the voices of the cosmos”. I’ll just quote the highlighted part from my original post:
Originally, with regard to the secular reasoning:
…the organ was supposed to be the combination of all the voices of the cosmos. Accordingly, the organ music at imperial utterances meant that when the divine emperor spoke, the entire universe resounded.
How the Church began to view it:
The “organon” is the cosmic instrument and as such the voice of the world’s ruler, the imperator.18 As against this Byzantine custom, Rome stressed a cosmic Christology and on that basis the cosmis function of Christ’s Vicar on earth: what was good enough for the Emperor was quite good enough for the Pope.
and… how the secular surroundings, such as the emperor, was against the Church and the Pope having “cosmic claim”, etc… and how the Church fought this by keeping the use of organ basically in order to put the emperor in his place:
To the exclusivity of an imperial theology which abandoned the Church to the Emperor and degraded the bishops to mere imperial functionaries,19 Rome opposed the Pope’s cosmic claim and with it the cosmic rank of belief in Christ, which is independent of and indeed superior to politics. Therefore the organ had to resound in the papal liturgy as well.
In addition, the use of the organ was to separate itself from the old ways of praising God in terms of music and other traditions. Although the Church has an inheritance from the Jewish faith, she wanted to be sure to not be like worship in the Temple, showing that the Church surpassed that of the worship in the synogogues due to its claim to the divine cosmos - the “very dimensions of the Universal”:
Such a borrowing from imperial theology is not regarded with favour by contemporary theological scholarship, which considers such acceptance as “Constantinian” or as “Romanisation,” which is naturally far worse than Hellenisation. As a matter of fact, what has been said thus far suffices to indicate clearly the convincing reasons for the whole process, as well as its logic within a Christian context: this detour made it possible to avoid turning the Church into a synagogue and to carry out in practise the true claim of the Christian faith, which accepts the inheritance of the Temple and surpasses it by far, into the very dimensions of the Universal.
For me, just based on this history alone, to use the argument of what was used for worship and praise in the Jewish faith during the OT, is no longer a good one. Prior to reading his Holiness’ article, I wouldn’t have had a good enough answer for myself, at least, to this argument except that this was what the Church wanted, not what Jewish faith wanted. But just this bit in his well-researched and interesting article/lecture on Church music, provides much explanation. The Church was supposed to have “surpassed” this, creating a new way of worship and prayer because she is supposed to have the fulfillment of the Truth through Christ. Whether or not people agree with that is one thing and can make decent arguments and very good reasons for using different instruments other than the organ, without using the OT/David used it, etc. argument. But based on this, there was obviously a lot of well-thought out reasons behind much of what the Church chose to do to delineate all of this, especially early on in the Church’s history.