A housemate of mine thinks that theories making testable predictions is unimportant, relative to how simple they are and what not-currently-testable predictions they make.
There’s some merit to this. There are testable theories that are bad/useless (luminiferous aether), and good/useful theories that aren’t really testable (the many-worlds interpretation of quantum physics). Goodness and testableness aren’t uncorrelated, but by rejecting untestable theories out of hand you are going to be excluding some useful and possibly even correct theories. If you have a compelling reason to use a theory and it matches well with past observations, your understanding may be better if you adopt it rather than set it aside to look for a testable one.
But there is a reason to keep the testable-prediction criterion anyway: it keeps you out of local optima. By the nature of untestability, a theory that does not make testable predictions, no matter how good, will never naturally improve. You may switch, if another theory looks even more compelling, but you will get no signal telling you that your current theory is not good enough.
By contrasts, even a weak theory with testable predictions is unstable. It provides means by which it can be shown wrong, with your search pushed out of the stable divot of “this theory works well” and back to searching. If your tests are useful, they will push you along a gradient toward a better area of theory-space to look in, but at the least you will know you need to be looking.
The upshot is this: even if you have a theory that looks very good, in the long run it is probably better to operate with a theory that looks less good but has testable predictions. The good but stable theory will probably outlive its welcome, while the testable but weak theory will tell you to move on when your data and new experiences pass it. Like a machine learner adding random noise to avoid being stuck, testable predictions are signals that ensure you will explore the possibilities.