Plenty of progressively-minded people are responsible.
Progressives take personal responsibility, from my perspective, only in so far as they do not behave progressively. For instance, you behave in personally responsibility-taking way yourself: You identify a career path, you take actions to pursue that career path, and you fulfill duties that correspond to the positions along that career path. You derive meaning from this at each step of the process, and it is an intensely individual project. I see the antithesis of progressive politics in this line of behavior.
As the counter example: Someone who does not identify a career path, who does not provide for themselves, who takes on little to no personal responsibility, winds up unhappy and in a poor SES position. For this person, the political progressive recommends more not-taking-on-of-responsibility, because "it isn't their fault". Protests do not amount to taking personal responsibility practically by definition. However, that's only the easy half of the picture to caricature, and this isn't necessarily even Peterson's target audience, because those people have for too long fallen victim to the idea of rights without responsibility. Instead, he is talking to those people who may be muddling through the bare minimum, but without purpose. College students who are there simply because it's what you do, with an attitude of "Cs get degrees". As a male, he's particularly challenging males to take on responsibility for themselves, in society which has a pathological aversion to it. We are supposed to take responsibility for the actions of dead others, but not our own actions and futures.
In response to the other things related to poor writing, mythological history, and rudimentary literary analysis: I haven't read Maps of Meaning straight through or even significant excerpts, because it is a sprawling text (and probably poorly written) and I'm not all that interested in the detailed connecting threads of mythology across culture and time, other than a cliffnotes format saying "hey look at these continuous themes". It's true that there are old value judgments in these mythologies, but Peterson's point isn't that we can or should return to these specifically, but rather that we need to be aware of how these inform our presuppositions and society, and how they derive from us biologically to some degree. This is the same reason he made the point about lobsters in the Channel 4 video. Hierarchy, sex differences, etc are in our DNA and go back much farther than our verbal or written mythos. The degree to which societies fail are in the degree to which they ignore these things. To put it in vulgar terms, The New Soviet Man starved to death.
