If you have a theory that is impossible to falsify experimentally, we're no longer talking about science (at least if your idea of science is the pursuit of objective truth). Modern "science" running amok with people more concerned with attempting to prove themselves correct than find the truth doesn't change that.
> Popper's falsification is a pretty unrealistic account of how science actually works
That's the rub though. A pretty strong argument can be made that "science" is not working and that "scientists" have lost their way, if you look at things like the reproducibility crisis.
Even assuming what you say is true, if not experiments and attempts at falsifying your ideas, how does one pursue the truth?
How would you suggest we run randomised experiments on the climate (or the macro-economy)?
They have most of the same issues as economics (modulo the political stuff), but everyone seems OK with it.
Also, Popper's falsification is a pretty unrealistic account of how science actually works, and it only really works as advertised where you can run effective experimental designs (which economics and geology, amongst other disciplines) have a problem with.