Why does a physical theory need to be testable? We are generally not interested in physical theories that cannot be tested with the scientific method. This would seemingly apply even if the theory has other advantages, e.g. simpler, more general, more elegant. Of course the possibility exists that the "correct" theory may not be testable beyond what it is designed to explain, or may contain many untestable consequences. In this case we would be eliminating such a theory a priori. Isn't this a problem?
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may not be testable beyond what it is coming to explain, or may contain many untestable consequences. In this case we would be eliminating such a theory a priori. Isn't this a problem?

That’s all well and good. Until you have a plethora of such “theories” to choose from. Which one do we pick? Which one do we fund?
If it is consistent with current theories and observations, and it can be shown to be equivalent to existing ones, it becomes another formalism. For example, the path integral formalism of quantum mechanics.
The scientific method is stringent on the falsifiability and that’s where it gets its strength from. That’s how we can safely pick one theory over the rest by seeing which theory best predicts what we observe. If we loosen this, then we lose the strength of the scientific method.
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This would seemingly apply even if the theory has other advantages,
e.g. simpler, more general, (more elegant?)

I beg to differ. If it is simpler or more general, it will be picked up sooner or later as main theory. If it is more elegant but not simpler nor more general, it will not get adopted in practice, but many people will study it just because of its elegance (albeit it seems to me the elegance is in physics equivalent to simplicity and generality, so "more elegant but less simple and general" seems to me as oxymoron). We are also not at the end of scientific investigation and the elegant ideas might prove to be useful for future theories.
A: Theorem. $\;$
A physical theory does not need to be testable.
Proof (by counterexample). $\;$
There has been a strong interest for string theory but it is not testable because of theoretical and mathematical difficulties and  because of the extremely high energies needed. $\; \Box$
Proposition. $\;$ String theory is not a mere mathematical theory.
Proof. $\;$
It is testable in principle (although not in practice). $\; \Box$
A: One obvious example of a "simpler, more general, more elegant" theory is that the big bang was initiated by a supernatural being that created the universe for a specific purpose, which we have so far been unable to discover.

In this case we would be eliminating such a theory a priori.
Isn't this a problem?

Is it "a problem" if science completely rejects this perhaps true theory and searches for different, but possibly testable, explanations?
Most scientists don't find it a problem to reject the concept of God.
Even those that do hold personal religious beliefs tend to keep them separate from their work.
That's how science works, and so far it doesn't seem to be a problem.
