Psychological depth in Tolkien's characters

[The Saucepan Man]

I think that the psychological depth, and the hunt for the readers to find it within certain characters, can be a mystery for us.

If you are talking about "implied psychological depth" in the sense that a character's psychological depth may be implied from his or her actions and interactions, then I agree with you. I think that Eowyn and Denethor are both good examples of characters whose psychological depth is implied by their deeds and by their dialogue with other characters. But, if you are talking about characters whose depth we have to fill in with our own imagination, then I disagree. How can a character, as written, have psychological depth if we have have to fill in the gaps ourselves? We may imbue them ourselves with psychological deoth, but that is a different matter.

I think they do have to have psychological depth so that we, the readers, will find the characters interesting.

No, there are a multitude of reasons why we might find characters interesting and care about them. Psychological depth is only one such reason. Heck, characters don't even need to have particularly great characterisation for us to care about them if they are important enough to the story or sufficently well integrated into it. I think that it would be difficult to argue that Legolas and Gimli are particularly well-developed characters, whether psychologically or otherwise. And yet you would presumably say that you cared about what happened to them. Why?