Psalms can be used as prayer - when you want to pray and don’t have the “words.” It can be a script which can be used to jump-start your soul. It is a shared spiritual language, that can be used to build community: a Psalm circle in which the Book of Psalms is assigned out to a number of people - but one person can use the Psalms on their own - reading anything from one phrase or saying one Psalm, several, or many - or the whole book.
Psalms are “extra curricular” prayer. To be sure, Psalms are found all over our set liturgy, but even if a person says our required, regular, daily prayers, the Psalms provide an opportunity to do a bit “extra.”
I’m not suggesting that saying Psalms has a cause/ effect relation with cure - any more than any other sort of prayer. It isn’t a magical incantation. We can’t manipulate God. But it can be centering - “mindfulness.” And while there is certainly no scientific data showing that it “works,” (causing a cure), there is a *ton* of sociological data that it “works,” (helps with coping) — the evidence being that people have been doing it for centuries. If it didn’t bring comfort, people wouldn’t do it.
If you have trouble with “belief,” that’s fine. You aren’t the first and everybody has doubts. Healthy doubt is to be expected and has a place in spirituality. But don’t be too skeptical! Yes, there is naive belief, but there is also naive unbelief.
Contemplate that all societies - including secular ones, have poetry and have ceremonials. This is ours. Try it.