Dear Emily,
I think your essay digs around some questions that have become crucial in physics this century, and can be seen as part of a self-searching process that science is now engaged in. To some, it's an identity crisis, but however one takes it, it's very helpful to shine a searchlight on these questions, as you do.
I also tend to take an overview kind of position, standing further back, as you do - to me two of the strands you mention need to go hand in hand: conceptual understanding, and the empirical side, mathematical theory/experiment. I agree that there's more to conceptual understanding that just a reverse form of prediction, though it's a good point nonetheless (you mention in a footnote that identifying emergence is another way we sometimes reach understanding).
To me, with a jigsaw for a loose analogy, theory and experiment tells us how the pieces fit together, but conceptual understanding - seeing the picture - is also essential. They need to work together, each informing the other.
I think the reason we're having to question our aims now is that one of these strands got left behind during the 20th century - theory/experiment got ahead. So the conceptual side needs to catch up, and we need to interpret the mathematical theory that we have.
The main point in my essay is that to interpret QM, missing concepts must be found. Several things suggest missing concepts, including the lack of consensus on how to interpret the theory. But also the physics itself - apparently fundamental unpredictability, non-locality, superposition. If there are missing concepts that encompass these, we'll need to find them.
Anyway, congratulations on another good essay - I hope you'll read mine, which is here. As well as looking at the overview and the different strands, it summarises an entirely new interpretation for QM, which anyone interested in quantum foundations might find interesting. A documentary was made partly about the interpretation last year, with some well-known physicists discussing it. I'd very much like to hear any thoughts you might have on my essay.
Thank you, best regards,
Jonathan