I don't have time to make the biblical case, since it requries a systematic understanding of the nature of OT worship and the changes into the NT. I believe it was offered as a "compelling thought" rather than an argument anyway. To leave some more pieces in this way of thinking, the psalms need to be understood as prophetic literature. David is viewed as prophet in the OT and the NT. The NT often makes use of their prophetic nature for the instruction of the church. If one views the psalms with an eye to prophecy, one will notice the psalms have a universal call to praise the Lord. This call grows as the books within the psalms move forward to the end point of psalm 150. As with all prophecy, there are NT realities described in terms of OT institutions (such as in Isaiah; note that the usage of the psalms in the NT show that they speak from the perspective of the NT, "today"), but the point of the prophecy is to describe the NT realities.
When this is understood, one will notice that psalm 98 calls the Gentiles to praise with harps, trumpets, etc. One will notice psalm 150 calls everything that has breath to praise the Lord with a variety of instruments. How are the Gentiles supposed to do this? These instruments were appointed by David for Levites and not for the common person to take up and use in the worship assembly. How will they join this Levitical choir and instrumentation? And these instruments, being attached to the temple, have long since disappeared. This suggests that the instruments here are speaking of the praise of the Gentiles, which we know continues. Yes, there were some Gentiles under the OT praising the Lord, etc., but this must be looked at redemptive-historically, since psalms such as 98 are looking toward that time when the Gentiles are brought into Israel to praise the Lord.
Other lines of evidence that lead to this. 1 Chronicles 16 (already mentioned) has the Davidic instruments set up in connection with a psalm that calls to the Gentiles. Even some pro-instrument (as in, instruments as an element) recognize the connection between the orcestration of the instruments and the calling of the Gentiles to praise the Lord. Almost every psalm (potentially all of them; I have not finished studying this part yet) that mentions the use of Davidic instruments has some connection to the calling of the Gentiles, e.g., psalm 108 mentions David praising the Lord among the nations upon the heels of praising the Lord with the psaltery and harp. Psalm 33 has a call for the earth to fear the lord, after telling the righteous to praise the Lord with an harp. Furthermore, in Revelation where all OT imagery returns, we see harps return in connection with the praise of the saints. As incense symbolized their prayers, harps symbolize their praise; confirming the association of OT Davidic instruments with praise under the NT and so the manner in which the Gentiles could be said to praise the Lord with an harp in psalm 98 (indeed, in Revelation 5 we note again a reference to redemption from all nations). Romans 15 connects the praise of Christ among the nations to various OT psalms, some including mentions of praise with Davidic instruments (and has the Gentiles praising with one voice with the rest of God's people, even as the instruments were said to do in Chronicles). Hebrews 13 shows that the sacrifice of praise is to be made with the fruit of lips, rather than with Davidic instruments, further suggesting their association with praise.
Questions as to why these instruments were not used before seem to me irrelevant to the question as to what they represented once they were instituted. But although I am not sure exactly what's up with the trumpets, it is clear that they represented something terrifying in the Mosaic institution but became associated with joy in the Davidic institution; perhaps a representation of how the law is sweetened under the gospel (for though the Mosaic ritual had sacrifice, etc., the gospel is more clearly seen under the Davidic ritual given the addition of a service of song with songs that call the Gentiles to praise)? Also, it is clear the Davidic institution is an embellishment of the previous one: the trumpets remain and more instruments are added as the Levites who had one task are now needing another task. It should also be noted that there is a progressive nature to revelation, so I do not see why everything that the Davidic instruments represented ought to have been present in the Mosaic worship ritual. If one feels one must answer the question: it would seem the Lord was pleased in the progress of revelation to await the Davidic institution, especially given the connection between David and Christ in the NT, since Christ is the key to the calling of the Gentiles (Romans 15). Given how much the calling of the Gentiles occurs in jubilant psalms, it also seems fitting that this part of the progress of revelation (i.e., including Gentile praise in the stated worship; the mystery of the Gentiles was revealed earlier, of course) would occur with the institution of a service of song to praise the Lord.
There are still some minor difficulties to sort out (to my own mind; I'm still thinking about this/studying this), but they do not seem to me to be insuperable. It is hard to ignore the evidence of the connection of the Davidic instruments with the calling of the Gentiles, however other details fit into this evidence.
in my opinion (seriously, just opining/wondering), those who are pro-instrument and view instruments as an element of worship or as something circumstantial to the OT worship will likely not see this as a compelling thought because of a different systematic understanding of the OT worship rituals, etc. But those who are anti-instrument in worship will likely find this a compelling thought (although no careful argument has been made) because the process used to arrive at the view will have already started incorporating that systematic understanding needed to see it.
To clarify the acapella position, it is not necessary to figure out what exactly the instruments represented to know they were part of the ceremonial worship system, as with any other ceremonial worship practice. It is clear from the OT that they were especially incorporated into its worship and associated with sacrifice. It is clear they came with specific instructions for their use that cannot be carried out today. It is clear from the NT that worshipping God by "elements of the world," by "things that are made," or by "carnal ordinances" are what constitutes ceremonial worship. Instruments are material objects and things that are made and were used to worship God. The NT also classifying instruments as "lifeless things" puts beyond doubt that the instruments were part of those carnal ordinances, the NT being characterized by living stones and calling for living voices (and only worshipping God by two material ordinances: baptism and the Lord's supper). Hebrews also points out we have not come to the sound of a trump, showing the ceremonial usage of the instruments that we no longer are to have (the trumpets were in place to symbolize the sound of the trump on Sinai; the other instruments were embellished upon this; since NT reality is something other than Sinai, this shows the instruments belonged to a former dispensation). And so forth (e.g., the arguments from Hebrews 13 and Ephesians/Colossians given by Rev. Barnes above).
(Of course, in the OT, the worship ordinances were simply all there together. We cannot determine what was or was not ceremonial without instruction from the NT as to what constitutes ceremonial worship. Having consulted the NT, we can then also see indications in the OT of the ceremonial nature of some ordinances, e.g., how praise is preferred to sacrifice shows its non-ceremonial nature.)
