Suppose we build a universal quantum computer and it functions as we currently expect it to. Suppose, further, that this universal quantum computer is of a nontrivial scale (hundreds+ of logical qubits), and we accumulate enough evidence to make a strong scientific case that the only thing that stops us from enlarging that scale (and thereby having more qubits, more gates, and longer runtimes) is engineering details that can be incrementally improved on.
What does this tell us? In a nutshell:
This is a simple statement, but I mean something very precise by it.
By quantum mechanics, I mean QM as a mathematical theory used to explain the world and predict the outcomes of experiments. This includes the unitary evolution of isolated systems, as well as the probabilistic behaviour encapsulated by the measurement problem. This last bit is necessary, since you've stipulated that this quantum computer 'functions as we currently expect it to', which is fundamentally based on the probabilistic read-outs enforced by QM.
By 'works', I mean that it is a correct and useful framework within which one can predict the outcomes of experiments and design interesting new experiments, as well as a useful framework for understanding what's "really" going on in the experiment, however (un)satisfactory that understanding may be.
I also mean that the extent to which QM works would be demonstrably larger than it currently is. Today we think there's no physical principle that stops quantum mechanics from working on the highly entangled, information-rich environment inside a quantum computer. A working quantum computer would demonstrate that this is the case.
If you will, a working quantum computer would take us from where we are now ("QM works") to something like "QM works, we're just a little bit more sure of it now.
Most importantly, though, because a working quantum computer, as we currently envision them, sits completely inside quantum mechanics, it has no bearing whatsoever on the interpretations of quantum mechanics. The interpretations of QM sit entirely outside it: they are ways for humans to make sense of the mathematical formalism, and to try and explain the parts of QM which are inconsistent or undefined.
(Brief comment for clarity. Inconsistency as in "evolution is unitary. Also, evolution is probabilistic". Undefiniteness as in "systems smaller than about yea big follow unitary evolution, and systems bigger than yea big follow probabilistic evolution. What do you mean, how big is yea big?".)
You touch upon this issue in your question:
All of the mainstream interpretations of QM return the same predicted results, but differ in their explanations 'why'.
A working quantum computer would simply provide the 'predicted results', and we would remain as baffled as before as to 'why'.
On the other hand, if the quest to build a quantum computer turns up experiments that cannot be explained by quantum mechanics, then there is definitely good business for science. It would mean that there is a bigger theory than QM, and it would give us a starting place of where to look for bigger explanations, which will hopefully be less confusing. In that case, we would be able to 'interpret' QM from the outside but within a bigger theory which we'd know to be right. However, as you might guess, few people are very confident that this will happen.