I agree, but I think the difference in usage is smaller than in English. "A human who loves is the happiest of all" or "A human who fried himself an escalope realised he was failing at it" don't sound very natural to me, (maybe a native speaker would like to correct me if I'm wrong), but they work fine if you use "Mensch" in German. My impression is that the "biological" connotation of "human" (i.e., as opposed to cat, robot, extraterrestrial) is a little bit more prominent than in "Mensch".
(The latter is one of Eugen Roth's "Ein Mensch ..." poems starting, "Ein Mensch, der sich ein Schnitzel briet, bemerkte, dass ihm das missriet")
You're correct that those sentences sound strange in English using "human". I suppose English would use "A man who..." in those literary examples, which unfortunately adds a gendered element that's not present in the German.