The thing about this, though, is that it only works if you get lucky and none of your nouns share a gender. I feel like that undermines the argument at least a bit.
I feel like I could come up with some examples if I had kept up with German after high school. I remember it being difficult for a year or two, then it seemed more helpful as we got into more complex language mechanics. In any case, German felt more consistent than English, and most of the words just felt right with one gender or another (and in speech you could usually get away with something between "the" and "duh" if you weren't sure about der/die/das).
They are generally going to sound good together because the word and the pronoun co-evolved. If they didn't sound good together, either the pronoun or the word would have changed. That isn't about the gender being right so much as just the sound, though.
There are other possibilities, but trying to apply logical rules seems pointless; the whole point of language -- what actually makes something a language -- is a completely arbitrary set of rules.