I believe the distinction between soft and hard science is mostly propaganda in order to discredit other disciplines as "not really proper science". the main difference between what we call "soft science" and "hard science" is imho the amount of variables that have to be considered. working with human subjects makes this amount of variables approach infinity very, very fast. which is why, say, doing a clinical trial is so incredibly expensive and extensive. many experiments from the hard sciences may be more "in a vacuum" compared to those from soft sciences, but are obviously still influenced by the researchers conducting the experiment and all the other people involved.
"soft sciences" have the same standards (or
should have, I can't speak for your favorite journal

in terms of methodology, experimental design, sampling, hypotheses, and even theoretical models (if we believe Kuhn anyway).
or, in other words, when physicists were deciding whether to stick with a modified newtonian model of science or adapt an Einsteinian one, their decisions was primarily an emotional, social one, and only to some degree based on evidence and reason. (sorry for the very rough generalization, you can read up on this in "the structure of scientific revolutions" if you care)
I have never seen an economic text with anything like the rigor that would get it published in a "real" science journal, and see no reason why considering the vast availability of economic data. My conclusion is that no economists care about truth as much as advocacy.
I agree with all of your post, but want to add this.
the fact that many studies in economics "lack rigor" as you state may also be a result of that scientific discipline having completely different standards than your discipline does. just as a quick generalization: in humanities a researcher is usually asked to examine his/her own biases, and not doing that would be akin to a grave mistake in many contexts. I doubt this is stressed as much in mathematics, although it may also play a crucial role there, too. that is just not how the discipline of mathematics has historically evolved, thus this criteria is not stressed as relevant.
what you say about economics and to an extent psychology is of course not wrong, I think it's simply not exactly the right diagnosis. psychology especially has struggled for years now with what is generally called "the crisis of reproduction", which is the fact that many gold-standard, influential studies are actually not at all reproducible. the fact that the discipline itself acknowledges this, and is trying to work against it or correct it, is imho a sign of a relatively healthy scientific establishment.
and this is where I would diagnose the problem with economics. it is not so much that economists, from our outsider perspective, lacks the scientific rigor (I doubt we're in a position to judge this fairly), it is much more than the problems that this discipline faces are all too oftentrivialized, explained away, ignored and so forth, instead of being dealt with in a healthy and productive matter. I thus call economics the ostrich discipline. while perhaps the humanities, if I may generalize, are almost too painfully self-aware of how they're perceived by other sciences or laymen, and as a result have made a conscious effort to appear more as a "hard science" (which ironically might not be the most productive way to research). (classical) economics as a science prefers to dig the head in the sand and talk to itself when criticism comes flying by.