No, I don’t think that’s something that’s normally done. If you want to speak in French without genders in that fashion, you basically have to talk without using pronouns or articles. And French likes those. Even if you’re only answering a question with a single noun, you’ll usually add some article in front of it.

What are you drinking? Herbal tea.
Que bois-tu? De la tisane. / Une tisane.

It would be possible for such answers to be given without the articles, but it would be very familiar, reaching to lazy.

Articles are generally not omitted except when talking about words themselves. For example in games (Scrabble, Des chiffres et des lettres, Mots croisés, …), one may enumerate as follows:

Une boisson en quatre lettres ? Lait, café, grog, saké, ouzo…

Pricelists is another example where articles are not used.

The only situation I can think of when not using an article is ok is when asking someone for something, with the reply:

— Tu veux boire quelque chose? Lait, café, thé?
— Café!!

As stated in other answers, this is used in familiar situations, and always in oral form (unless, of course, you’re writing a novel and your characters say stuff…).

French sentences are built a certain way: words go by groups so that you generally have groupes nominaux and groupes verbaux. A groupe nominal is normally composed this way: déterminant + nom.
For example in:

Les enfants aiment boire du lait

You have: Les enfants = groupe nominal (sujet) which is composed of a déterminant défini + nom, followed by aiment boire which is the groupe verbal and finally du lait = groupe nominal (objet) which is again composed of a déterminant indéfini + nom.

However, there are a few cases in which there is no déterminant. But then it is very grammatical:

• when the noun is an attribut du sujet : “Son père est médecin” (His/her father is a doctor)
• when it is apposé, i.e., when you want to specify something: “Monsieur Y, avocat de renom, a plaidé la cause de monsieur X” (Mr Y, the well-known lawyer, pleaded a case for Mr. X)
• when it is épithète: “ce fut une guerre éclair” (it was a blitzgrieg)
• when it is used as an apostrophe. The better example is this one: “Garçon, l’addition s’il vous plait” (waiter, the bill, please)
• when it comes after a preposition: “j’aime me promener sans but” (I like going for a walk, even with no particular destination)
• when it is a kind of definition: “Lune est de genre féminin” (weird translation in English as nouns are not gendered: moon is a feminine word)
• in locution verbales: “avoir tort” (to be wrong), “demander pardon” (to ask to be forgiven)
• in expressions figées, i.e, idiomatic sentences: “il y a anguille sous roche” (equivalent of “there is something wrong”) or “pierre qui roule n’amasse pas mousse”)
• in notices or signs: “maison à vendre” (house to sell) or the very old “école communale” (school).

Hope this can answer your question. By the way, I wouldn’t have been able to give you this quite complete list of exceptions without the help of my grammatical Bescherelle.

You’re thinking of the null article in English, which is quite different from an elided or omitted article. It is not a nothing, but a something that has a quite specific meaning: “any” or “all.” If you want a sentence like The boys chase this girl a lot. to have a more global meaning, you can say Boys chase this girl a lot. The first sentence characterizes some specified boys; the second is all about the girl.

You can do the same thing in French, but not with a null article, for very good reasons. Consider: this is already a language with the final consonants mostly elided, or peeled off and affixed to their following vowels, which I think is the point you’re making when contrasting “milk” and “mil.” This French penchant for elision and liason creates a musical stream of consonant-vowel syllabications well suited to diplomacy and girl-chasing, but it also clobbers noun inflections that identify number and gender. The articles that also carry these inflections are therefore not dispensible.

When you cross someone in the street and say:

— Salut !

He responds,

— Salut

And you both keep going your own separate ways. That’s approximately how long a gender-neutral “conversation” lasts in French. Gender is just part of the language and there is no avoiding it really. There might be genderless sentences here and there, but that is about it.

In French, gender is not marked directly on the nouns but rather on their satellites (determiners, adjectives, participles). Some determiners and many adjectives do not mark gender such as l’, les, des, calme, tranquille, etc.

Some other determiners do not mark gender in a straightforward way and can be misleading when a vowel initial word follows them :

• mon amoureuse (F) : mon sounds like the masculine possessive
• cet avion (M) : cet sounds like the feminine demonstrative

Some French speakers have a hard time figuring or remembering the gender of vowel initial nouns and usually end up arguing about these things (e.g. augure, interface).

Some nouns like après-midi might even have two genders with a different complex distribution. Après-midi (M) and après-midi (F) being the rough equivalents of jour/journée, an/année…

So noun gender is not always obvious to French speakers and in many cases even when it is, the marks are not always readily available (or audible) on all the satellites :

• Les jeunes journalistes sont encore arrivées en retard.