I think you may have something there. Seriously! Psalm 23 is usually sung to a repeating tune. Keith Green totally changed that with his musical score - he changed the music and (In my humble opinion) breathed life into it. I can't sing it now without thinking of his arrangement.
Strictly speaking Psalm 53 is beyond the remit of this commentary but given the similarity to Psalm 14 I think something needs to be said. The psalms are not identical twins but are almost the same. We could of course focus on the similarities and speculate about changes to the original. I would prefer to focus on the differences first then discuss this second version of Psalm 14.
Leupold suggests two alternative theories. The first is that a scribe discovered a defective/imperfect copy of Psalm 14 and attempted to restore it. This sounds like what we would do today but does not accurately reflect the care and respect given to copying the text. The second is that someone used Ps 14 as a template to write a new Psalm.
Verse, Psalm 14, Psalm 53
They have all turned aside
Every one of them has turned aside
Do all the workers of wickedness not know, who eat up my people as they eat bread, and do not call upon the Lord?
Have the workers of wickedness no knowledge, who eat up My people as though they ate bread, and have not called upon God?
There they are in great dread, for God is with the righteous generation
There they are in great fear where no fear has been; for God scattered the bones of him who encamped against you; You put them to shame because God has rejected them
You put to shame the counsel of the afflicted, but the LORD is his refuge
O that the salvation of Israel would come out of Zion! When God restores His captive people, let Jacob rejoice, let Israel be glad
Oh that the salvation of Israel would come out of Zion! When the LORD restores His captive people, Jacob will rejoice and Israel will be glad
I would like to suggest a third option. It is a Psalm of David in which he wrote in a slightly different vein and for a slightly different purpose. Having written a little poetry I often leave a trail off earlier versions, sometimes coming back to them. I suspect that just as there may be different versions of a song. Perhaps the best example (or most contemporary) would be the rewriting of "Like a Candle in the Wind" by Elton John to commemorate the life of Princess Diana.
Psalm 14 uses Jehovah four times and Elohim three times. Psalm 53 uses Elohim seven times. Psalm 14 seems to deal with internal Jewish problems and deals with a righteous generation within Israel. Psalm 53 seems to deal with an external enemy that Israel defeats in battle, aided by God. This would be my starting place for distinguishing Psalm 53 as distinct but repurposed.