What is particular about Psalms according to Exclusive Psalmodists?

The denomination I have been with for over two decades were, when I began association, exclusive Psalm singers for worship, and rejected all musical accompaniment, having a lead singer at the front to pick the tune and the key. There was a lot of spontaneous harmonising of voices, and it was a very dignified way of conducting sung praise. I soon entered into it, despite having previously been a key-board player at a different denomination's worship services.

However, since 2011, all congregations could decide for themselves whether they wanted to introduce hymns and other praise songs, and musical accompaniment also, but they had to maintain some unaccompanied Psalm singing. Some congregations stuck to the status quo but quite a few others (our own included) agreed to introduce those additional forms of praise. So, I found myself back at the keyboard for some parts of the musical worship!

Details from the official Statement, following deliberation of its Plenary Assembly in November 2010, provide information as to the previous stance taken in 1905. It was only then that men taking ordination vows (to sign The Formula) had to include one about unaccompanied Psalm singing. However, they were not bound to all the regulations as set forth in the Directory for Public Worship, or the Form of Church Government. They were bound to maintain the whole doctrine contained in the Westminster Confession of Faith: "The Confession alone contains the Creed to which every office-bearer 'must testify in solemn form his personal adherence'.

This is an interesting point due to this statement in a Minister's Article, May 6th, 2011: "'Ordination vows' have been invoked to proscribe all appeal to the Bible." This article goes on to explain, re. sung worship in congreagations:

"We are bound in conscience to the Regulative Principle, which John Calvin in his tract, The Necessity of Reforming the Church defined as follows: 'God disapproves of all modes of worship not expressly sanctioned by His Word'. This Principle is an integral element in the Confessional commitment of every Free Church office-bearer: God may not be worshipped in any way not prescribed in holy scripture (Confession of Faith, Chapter XXI:I).

All of us are bound to it. The only question is, How are we to apply it in practice? Calvin and his successors are much clearer on the negative side of this principle than they are on the positive. Things like the use of images and the worship of saints are forbidden. But when it comes to working out what is prescribed, we are very short on detail because the New Testament gives us little information on the worship of the apostolic church. On the face of things, the singing of hymns and spiritual songs, far from being forbidden, is prescribed (Ephesians 5:19), but opinions will differ on this, of course.

This confirms Cunningham's observation (The Reformers and the Theology of the Reformation p. 32) that we have to interpret the Regulative Principle in a common sense way, allowing for difficulties and differences of opinion over matters of detail. This is already hinted at in the terms of the Formula. Our form of worship is 'founded' upon the word of God. It is not derived from it in all its details...

The 1932 Act anent Requirements at Ordinations and Inductions sought to regulate the Church's sung praise, forbidding the singing of 'uninspired materials', and precisely because it was an Assembly regulation, even those of us who did not believe in exclusive psalmody felt bound in conscience to comply with it. All of these regulations have the authority of the General Assembly, though some are widely disregarded (for example, the 1910 Act anent Postures in Public Worship)...

We are bound in conscience to comply with the terms of the Act of the Plenary Assembly of 2010 [which allowed Kirk Sessions liberty to introduce into public worship hymns and musical instruments, if they so choose, (alongside Psalms)]... Every minister is now bound by this: to recognise the right of each local church to decide whether or not to sing hymns as well as the cannonical psalms."

The history of all the developments regarding sung worship in congregations is shown by various Acts then repealed by the General Assembly in November 2010, in order to enable this greater freedom to obtain. They repealed 10 Acts and resolutions from 1872 through to 1932.

Now, although this does not give an Exclusive Psalmodist's arguments FOR such exclusivity, there is something to be gleaned from looking at a formerly exclusive stance in one denomination, to see why they came to see that their previous stance needed to change, and there are some key phrases in the quotations that may indicate why there has not yet been an answer from an Exclusive Psalmodist. However, I hope this answer might prompt the kind of submission being looked for. This answer really deals with many questions / issues raise in the comments.