As a native English speaker, grammatical gender seems unnecessary. Does it provide nuance or clarification of meaning?
Interesting question, so I looked it up.
Wikipedia offers these, but I find them to be a stretch. It claims #2 is the most important, but in a language with 2 genders, you only have a 50/50 chance of it helping.
I was thinking more about the perspective of a native speaker of a grammatical-gender language in every-day speech. It seems weird to me to think of a table as either masculine or feminine… I agree the wikipedia examples seem contrived.
I don’t think native speakers of gendered languages think of an object like a chair as “masculine” (as in the German “Der Stuhl”) or “feminine” (as in the French “La Chaise”) in the way you’re thinking of - it’s just a grammatical property that goes along with the word (the French word for “beard” is feminine, the French word for “vagina” is masculine.)
I do not speak German but I believe that the diminutive in German is always masculine. Fraulein is a diminutive for Frau, so contrary to intuition is masculine. So the words “masculine” and “feminine” to describe grammatical gender are a convention and do not literally mean the same thing they do when applied to people.
However, none of that helps to answer the OP, which is an interesting linguistic question. I also don’t buy the Wikipedia explanation.
Another point that doesn’t address the OP’s question directly but that may be of interest, Old English was gendered (in a manner similar to that of modern German.)
Not quite: German nouns ending in -chen and -lein are always neuter, so that Mädchen (girl) and Fräulein (unmarried woman) are neuter, even though they refer to female people.
Keep in mind that “gender” is a term linguist use to refer to a grammatical aspect. It is not meant to indicate that the nouns are actually female or male.
A fair number of the personal pronouns in English provide clarification by using grammatical gender.
I think it’s like with a lot of grammatical components / cases, where you can think of examples where it might resolve ambiguity, but is more often simply redundant.
In English, for example, is it useful to refer to a “herd” of cattle, a “school” of fish etc instead of some generic “crowd” word? No, not really.
(I’m sure there are better examples, this is just the one that popped out my head first)
But proper nouns and pronouns referring to women are grammatically feminine, and those referring to men are grammatically masculine, right?
No, not necessarily. As already pointed out, in German Madchen and Fraulein are grammatically neuter, not feminine.
Since “aspect” itself a grammatical term, perhaps we should say that gender is a grammatical FEATURE. 
When people hunted and depended on it for food, it was useful to have words for groups of hunted animals, and for farmers to have words for groups of livestock animals that allowed them shortcuts. “There’s a herd,” as opposed to “there’s a group of deer.” It doesn’t seem important to us, because we don’t say it often, but think of having to say “gigabyte” all the time, instead of “GB.”
A writer even spoofed it once, in the 15th century, and came up with lots of funny names, like “a superfluity of nuns,” “a fighting of thieves,” and “a glaring of cats.” Some of the names stuck, and a group of crows is now referred to as a “murder,” because of her joke. For all I know, “pack” of dogs may even have come from this source, although, it probably comes from wolves, and it does seem useful to have a special word for a group of wolves. Anyway, this writer did give us “gaggle” of geese, school of fish, and swarm of flies, all of which the invention by the author is unknown; they may have been current terms, or may not, but they are now.
I actually forgot my main point: gendered words go with certain adjective endings, and certain articles, and in inflected languages, certain case endings. They evolved that way because they “sound right,” somehow, the way a/an in front of some words and not others sounds right to us. That’s a klutzy sentence, but I think you get my meaning. Linguists chose to call them genders, and some languages have three, but they really have nothing to do with gender. We have a few gendered words, like actor/actress, which survive mainly because it’s one of the few jobs that can actually (and legally) call for a person of a specific sex. We’ve lost gendered words in a lot of other places, like waiter/ess-- now “servers,” steward/ess-- now “flight attendants,” and soforth. And women are governors of states, while men who live in and care for children are generally not called govenesses, but neither are women, usually, anymore; I haven’t heard the -tor/-trix distinction anywhere but the bondage community, which still likes a dominatrix. Women who carry out wills are executors, however. And the last time “Jewess” was in print, outside of some anti-Semitic thing (I don’t know why) was around the mid-1930s.
I say “gig” 
Your point about it potentially being useful to say “There’s a herd” makes sense, but I don’t think it’s necessarily the case. I think if we’re going back far enough in the language it would be acceptable just to say “There are cows”, with group words only coming into play when you need to refer to multiple groups, in which case terseness is unlikely to be that much of an issue.
Anyway, my original point stands because as you mention, many of our current group words were just invented one day for fun. It’s fun to say “a murder of crows” but there’s no real reason to have a specific word vs flock (or even scratch flock, just have group).
I like “aviatrix”, but know I’m being deliberately antiquated.
It seems like languages like English and Dutch (to a lesser degree) have lost a lot of grammatical complexity that a language like German has retained.
What I wonder about is where all that complexity came from in the first place. Did people who lived before we started writing things down really use this complex grammar so very consistently? That seems hard to believe.
It only seems complex to English speakers because English speakers aren’t used to it. Grammatical complexity in other languages does a good deal of the heavy lifting that, in English, is done by e.g. word order. In English, you know from word order whether a word is a subject or an object, or which noun is qualified by which adejective. In other languages, you might know this from the grammatical form of the words, meaning that those other languages can be much freer with word order. Speakers of those other languages might find English grammar comparatively easily, but English rules about syntax and word order massively complicated.
IIRC a number of languages have been through phases of having more complex conjugation and grammatical particles, and then simplifying (but simplifying conjugation makes things like word order matter more).
I can’t remember where I heard this, so could be complete BS…