As the title says, I would like to understand chemically why this occurs. My theory behind it is that in the former case both are completely dissociated, and so at the equivalence point (and approaching it), everything is controlled increasingly by the autoionization of water, whereas in weak acids there is also a much more sizable contribution of $H^+$ by the acid itself, so the change is not as drastic. If this is correct, I'd like further explanation as to why it is, since I don't quite feel comfortable enough with it intuitively.
This is not a duplicate. I am not asking why the strong acid curve is steep, but rather why the strong acid curve is steep while the weak acid one is not.!