Noun gender might conceivably work that way some of the time, for example if there are nouns that otherwise sound fairly similar but are distinguished by their genders; if you heard an article or an adjective that agreed with the noun, you could get an extra clue about what it was.

An example that I thought of in Portuguese is ato/ata, two closely etymologically-related words with different grammatical genders. The former means 'act, deed', while the latter means 'record, minutes (of a meeting)'. If there were a context in which these could be confused when spoken, surrounding words would likely reveal which one had been said.

I'm not sure whether this is commonly understood to be a likely reason that languages actually have systems of agreement.

Edit: another example of an agreement mechanism that people have to spend time studying is agreement of verbs with their subjects (in various ways depending on the language -- for example Hebrew and Arabic verbs agree with their subject in gender!). This is more complex and more effort to learn than in languages that don't do this or do it less, like Chinese and Swedish (which don't even mark person or number in the verb). You can imagine that this redundancy is sometimes helpful when you mishear either the subject pronoun or the verb itself, like the different between "we want" and "she wants" or "we know" and "she knows" -- if you didn't hear for sure whether the subject pronoun was "we" or "she", having it be fully or partially marked in the verb conjugation would help you figure out which it could have been.

So, to take an extreme, if a language was much harder to learn but much more feature rich for non-communication purpose (e.g. a better language to dream in), it could very well be much better to choose the complex language over the simpler one, even if the learning part is harder.

It's certainly not common, even amongst "PC" types or whatever, to want to eliminate gendered pronouns entirely; just that everyone gets to be called the pronoun of their choice (regardless of whatever genitals they were born with), and when the antecedent is of indeterminate gender, a pronoun of indeterminate or at least not a male-default gender is chosen. So Alice and Bob can still be "she" and "he" respectively if they so choose, and there's no confusion there; "they" still refers to both of them. But when a person wants to tell the story of Alice and Bob, that person is "they" or maybe "she" or sometimes "he" - just not "he" by default.

Anyway, slicing up our pronoun space into male/female is entirely arbitrary. Perhaps Alice and Bob are the people chosen for these kinds of stories because our pronouns are gendered; if our language categorized people in different ways than gender, we would chose people as examples along those lines instead. And what do we do when we have a story about Adam and Bob? They're both "he"! How ever do we deal with the combinatorial explosion of ambiguity? And yet people somehow have managed to tell stories with multiple men in them for millennia...

Suppose Adam is tall and Bob is short. Maybe English ought to have different pronouns for men of different heights (or hair colour, or...). It would help relieve this terrible ambiguity here. And yet we get by without this precision in our pronouns.

Perhaps Adam is black and Bob white. Should they have different pronouns to distinguish them? If you say no to that and feel like that maybe might even kind of be a bad idea because of the rigid line it draws between two people created equal, well, now you know why people might be opposed to gendered pronouns.

Some languages have the notion of politeness encoded in, which can often help distinguish pronouns in stories between children and adults, or royalty and their subjects. English doesn't really have this (at least not at a grammatical level), and yet somehow I don't feel my language to be particularly enfeebled without them - I can tell stories about little girls talking to their mothers, or male peasants talking to kings, just fine.

And as with "you", I suspect that if "they"'s use as a gender-neutral pronoun expands, "they all" will soon emerge as an (unofficial?) replacement for the plural.

Hm... Let me hijack the current state of languages the other way around: why would we have distinctions at all? Why would we need to express distinctively about animate and inanimate, or about the existing number of colors (when the entire spectrum can be botched to fewer hue names), or to make a distinction between single and non-single (English "you" FYI), or a lot of other stupid differences that force us to think about when we're communicating? Why wouldn't we drop them and market this as a feature, emphasizing the fact that we can speak that way "just fine"? Actually, you know what? Why wouldn't we drop communication entirely?! That would be a wonderful final goal, wouldn't it? I may sound like trolling, but it's important to see the whole picture - the goal of the communicating act in its essence is to transmit perceived distinctions. Now, we can shape the language used for this communication by pushing it either in one direction - of evolution (by gaining the ability to make even more distinctions), which is an open end indeed and that may seem overwhelming, or in the other direction - involution, which (hey, good for us -) has only so much to be reduced to!

Science (including the part addressing languages) is a bitch, isn't it? We have one less linguistic aspect available for politics now!