There is that tradition, yes, and it’s part of the history of the LOTH.
Jews had a similar practice in the Old Testament, and later Christian hermits and desert fathers would pray all the Psalms every day. That practice, though modified in various forms is found in the East and West. Kmon23 mentioned Eastern practice, which I was unfamiliar with.
In the west, since saying ALL the Psalms everyday is not easy or in many cases practical, it lead to the Liturgy of the Hours. The LOTH was essentially a modification by Saint Benedict, so the Psalms would be prayed and chanted over the course of several days. The Breviary most of us are familiar with is a modified, portable version of the large books monks chant from, and was developed/popularized by the early Franciscans so, as itinerants, they could more easily bring the psalter with them. (The early rosaries also began as a substitute for praying the Psalms, especially for the illiterate; hence 150 Hail Mary’s before the Luminous were added. ).
so…yes. What you are suggesting is exactly where LOTH came from. Although it’s not the “official prayer of the church” in that way, it’s still an excellent practice if you want to do that. I really love reading Psalms in the adoration chapel. I’m also interested in hearing what other people here do
I don’t know if you have a smart phone or not, but if you do, the “divine office” app, from divineoffice.org is excellent. It’s the same version found in Amercian breviaries. It’s a great way to access the LOTH when traveling if you have cell service.