I imagine a big appeal of falsifiability in its early days, stemmed from its opposition to religious dogma. Imagine dealing with someone whose answer to every question comes back to God. "Why did the patient die?" "Because God decreed it." Talking about germ-theory must have been a massive improvement over theology, hence why ideas like falsifiability must have arisen to provoke people to think in more critical ways.
My own standard for evaluating theories comes down to the following: The simplest explanation that best fits existing data, and makes useful predictions for the future.
"Simplest" because we need to prevent overfitting.
"Fits existing data" because the whole point of a theory is to explain what we see around us.
And "useful predictions" because it will actually provide practical benefit to us. As opposed to a theory like "God decreed it", which may be extremely simple and error-free, but provides us with no practical benefit at all.
As I see it, falsifiability is a side-effect of the second and third requirements. If a theory is making useful predictions, we might as well check to see if it's accurate. And if it's not, there's no point in holding on to a theory that's making faulty predictions. Falsifiability, in this perspective, isn't a requirement in and of itself - it's a natural consequence of the requirement that theories be useful and fit any data at hand.