This question is discussed by Stephen Kneale.

Do we look for Christ in all the psalms, or only in this customarily considered to be Messianic?

Expositors from Augustine to Luther tended to take the maximalist approach.  Calvin, however, stressed the original contexts of the psalms, and so adopted a more minimal interpretation.

In recent times, Bonhoeffer and Ash have leant towards the maximalist position, whereas others, including Dale Ralph Davis, incline towards a more limited reading.…

Fee and Stuart (How To Read The Bible For All Its Worth) outline three benefits of the Psalms for both their ancient and modern readers:-

1. ‘The psalms can serve as a guide to worship. By this we mean that the worshiper who seeks to praise God or to appeal to God or to remember God’s benefits can use the psalms as a formal means of expression of his or her thoughts and feelings.…

On a number of occasions, David and the other psalmists claim for themselves an innocence that appears to be unwarranted.  Was not David guilty of grievous sins, and did he not know it (Psa 51)?  Are we all not at best forgiven sinners, for whom protestations of innocence are always and entirely out of place?

This issue is helpfully discussed by Christopher Ash in Volume One of Teaching Psalms (chapter 6).  Here’s a summary.

We meet those who can be referred to as ‘righteous’ as early as Psalm 1:5.…