As a native speaker I can confirm the observations of Mark Twain. What is ignored in such humorous tirades is that every human speaker will push information density to the very limit of understandability. In every language. They will use all the tools available. Sure, the German case system is getting weaker as the centuries go by. And in some situations you will wonder why it's used still. But that doesn't mean the…
Why don't you think removing genders will make the language easier? I studied German in college and memorizing these was my only complaint.
E.g. female sun, male moon, female night, male day. Those examples might stem from codified personifications (e.g. nox).
In many cases though, I suppose it's a case of vowel harmony. If there was only one article, it would be "d'" with a semi-vowel ablaut, schwa, as in English "the". I don't know exactly why, I just recently heard of vowel harmony (ie. part of the native speaker's feeling for what sounds right) existing.
Gender has a minor advantage in some situations, I guess. I'm not sure whether the free word order is an advantage, but it helps for parsing this one: "Der Frau gibt der Briefträger einen Brief." - (to) the women gives the postman a letter.
Conversely, I find totally odd the neuter pronoun used on dogs. However, I do call all dogs "he" no matter the genitals, so that's no better, also because I don't exactly look for it. Whereas cat's are a priori female. It's not OK to neutralize (pun intended) a human person in any case.
Defaulting to "es", "they" or any alternative is still a crutch, as long as the inflection remains. And that has probably some reason to be there. Indo European had a two case gender system distinguishing animate and inanimate, female being a later innovation. In fact they had two words for fire, whence the ignition (animate) and the fire (inanimate). I do not know much about the case system, but taking Latin for example, that clearly got out of hand.
So, "die Flasche" (bottle) holds spirit. "der Wind" (wind) acts. But "das Wasser" just sits there waiting to be consumed. If it flows, it's "der Fluss". The sea is weird, "das Meer" is neuter, perhaps because as diminutive (Märchen, nightmare ...?), but "die See" is an abstractum (die Nordsee, rarely Nordmeer), whereas "der See" is any determined lake. Der Teich (pond), die Leiche (corpse), das Reich (realm), der Streich (trick) seem to resist the argument for vowel harmony, but if Streich and trick are from frankish or French, estrich, estriquer or sumsuch, then that's a different matter, same if Leiche was Lyke or so ... I don't know much about that.