Putting the OPs remarks to one side, that's a rather simplistic view of science.
> And the theories themselves can't be proven as true, since this isn't math but empirical science
This distinction is meaningful and extends to logic, metaphysics, etc.
> When people continue believing in theories that are non-falsifiable or that have been falsified, that's pseudo-science.
Popper distinguished between rationally justified claims and "scientific", i.e., empirically testable claims. He only claimed that falsifiability was a feature of empirically testable claims. That he should do so should be obvious because the claim that all claims must be empirically falsifiable is not itself empirically falsifiable and large swathes of human knowledge lay beyond the empirically testable, hence the restriction. That they are not empirically falsifiable does not make them pseudoscience. Indeed, the viability of empirical science hinges utterly on unfalsifiable assumptions.
> Scientific theories are a dime a dozen and in the presence of empirical evidence that falsifies them, we should consider such theories as falsified and move on.
Not so fast. Even within the domain of the empirical sciences, falsification is not simply a matter of performing a tidy critical experiment. As Duhem (and Quine) observed, the theory under scrutiny hinges on many other hypotheses, theories and assumptions. Even something as simple as observing something under a microscope hinges on the soundness of optics as well as many other assumptions that are not falsifiable.
There is most certainly an economy, a parsimony, an order to how science and common sense proceed when met with conflicting evidence. We tend to doubt the the outer layer of the epistemic onion where there is less certainty before asking questions about our assumptions.
And of course, science occurs in a social context such that it is subject to practical and cultural forces that influence what is given priority. In any case, the point is that falsification is not simply a matter of demonstrating that an expected outcome does not occur.
> but we can get an idea about how well they predict natural phenomena and we can disprove them by empirical evidence, observations that show these theories don't predict the universe well.
Predictability is only part of the story. If we reduce science to predictability, then we're back to a purely instrumental view of science. But as Hillary Putnam notes, science aspires to describe reality as it is. It does not merely want to produce a model that "works" which is to say one that allows physical phenomena to be predicated. In any case, that predictability should be a feature of empirically testable claims is not falsifiable, and predictability itself presumes that the universe is a certain way that affords predictability which is, again, not falsifiable.