I think I have an idea about what the reason might be. I found it on one of Qt’s pages. The libraries that are indirectly loaded from the directly loaded libraries don’t have the RUNPATH set, so they can’t find the indirectly loaded libraries since they are in a nonstandard location.
I could stick the RUNPATH in there with patchelf, but apparently chrpath won’t do the job. I’m stuck using LD_LIBRARY_PATH until I can figure out a way to get CMake to put it in there for me.