First of all, I believe the premise of your question is largely incorrect (and I know from your disclaimer that you are "not asking about that" but will take the liberty of addressing that anyway). At my university, researchers can use their research funds to pay for membership in academic societies, and my perception is that this is the norm in the United States. So, to the extent that it's not the case at your institution, my feeling is that that is out of line with the norm, so perhaps there is indeed no good reason for it and you should lobby to get this rule changed.
Second, even if it is the case that you cannot charge this expense to your research funds, the assertion that "academics are expected to pay out of their pockets for memberships in academic societies" still sounds highly questionable to me. What does "expected" here mean? I was not a member of any academic society until last year, and nonetheless managed to become a full professor and department chair at a respected university. So I maintain that no harm will come to your career if you ignore this so-called "expectation". Sure, it might be nice to be a member of a society and can come with various small benefits, but if you don't feel like paying for it, you'll be fine.
Third, even if it is the case that becoming a member of an academic society in your discipline is somehow a professional necessity, and that the only way to do that is to pay for it out of your personal money, I don't think that's necessarily unreasonable (nor is it entirely true that things aren't like that in industry - lawyers have to pay annual bar fees, for example, not all of which are covered by their employers). Your interpretation that this means "we are supposed to pay to do the job we were hired for" seems to me like a very narrow-minded view of what life in academia is about. Sure, it is a job and we are paid a salary for it; at the same time, being a researcher is much more than a job - it is a vocation, and that is why researchers are notoriously bad at separating their personal lives and their professional lives: most normal "workers" don't work late nights and weekends (not to mention holidays and family vacations) and don't spend a large proportion of their lives traveling for work, including moving repeatedly across large geographical distances before landing their first permanent job.
In fact, to be honest, as an academic, I feel like I can't draw a precise line separating my "job" from my "personal life". When I read a math book or article at the beach or swimming pool, am I "working"? When I think about a research problem while driving or talk about math with a friend over dinner, am I "working"? I honestly don't know. And that's a good thing. It means that I am doing something that makes me happy. How many other "workers" can say that? Too few, sadly. So let the industry people have their employers pay for them to do the "duties" that are "written in their contract", is what I say. If you are passionate about your work and being a member of a society is something that interests you enough, you would not find it burdensome having to pay a small amount for it. And if not, well, either switch careers, or just forego the society membership, and all will be well.