This is something I've thought about. I'm a physicist, but am now happily working in industry. My job is to solve problems, and for most part I do, but it always seems possible that one fine day, someone will walk into my cubicle with a problem that I just can't solve. I'm always faced with the fact that I can't honestly predict how long the next problem will take. Maybe a day, maybe a week, maybe... forever?
Despite a string of successes, we never know for sure when physics will be hit with a problem that takes a decade, or a century, or even a millennium, to solve. As I understand the history, the ancient Greeks could formulate quadratic equations that they couldn't solve, and it wasn't until something like a millennium later that Al-Khwarizmi cracked it, at least for real roots. Now we teach it to children in middle school.
Finding a testable theory of quantum gravity might be the problem that takes a millennium.
I think that I mostly agree with the theme of your post. An important thing to keep in mind is that physics is inherently an experimental science, so the critique of formulating untestable theories is valid. I have heard that cosmology is particularly bad about this, however I am not in that field...so who knows.
So instead you get all this hand-wringing about the process, about how theorists come up with elegant math that isn't testable, yada yada, without at least an honest statement that "It is at least possible that the universe's final secrets will never be amenable to human inquiry."