I frequently read the Bible. This includes the Old Testament. Recently I have been reading the wisdom books, such as Proverbs (which I find full of wisdom on everyday life, and helps me understand some things.) I find the Old Testament still quite relevant. I find the Old Testament still fulfilled the purposes of the Bible, which as Saint Paul said,“is profitable for teaching, for reproof, for correction, and training in righteousness.” (2 Tim 3:16). However, in some fundamental things the New Testament fulfills the Old Testament, so it can have some primacy.
I am not sure what your question is, though, since your heading in your opening post asks two questions and your opening post asks other questions.
However, many Christians like myself still find the Old Testament important and relevant.
And as someone mentioned, passages in the Old Testament are read at mass, the Catholic Church still finding the Old Testament important.