Because "xor" is not a word. Try using it in scrabble.

You are free to use whatever you want to prove theorems.

@markvs If you consider "iff" a word, then sure it is. Both are in the dictionary although xor is used as a conjunction here rather than as a noun.

There's an interesting question here (why are some connectives apparently less common in ordinary mathematical reasoning?), but I think this question is more of a philosophical question than a mathematical one. I also think it would benefit from focusing on a single connective.

@markvs What makes Scrabble the authority on what is a real word? Wiktionary does include xor as a conjunction.

@Okoyos: "Iff" is not a word either. That is why correctors weed out iff from books and papers.

@Sandejo: Scrabble exists much longer than Wiktionary (which is notorious for being full of errors).

@markvs Both xor and iff are in the OED. It is a word regardless of any suggestions by certain style guides or the opinions of "correctors".

@Sandejo: The older such a thing is, the more reliable it is. The most reliable is of course Oxford dictionary (oed). You can check if it includes xor or iff. There are similar dictionaries in French, German, Russian, etc.

@Okoyos: Oxford dictionary is of course reliable. But I would not put correctors in quotation marks. These people have real power.

@GregoryNisbet I feel it could be messy if people make a new post for every unpopular logical connective. I should make clear that an answer to this question only has to answer for one.

@Okoyos Counting true, false, and, or, implies, iff as commonly used, what makes you ask about xor in particular vs. the other $9$ less commonly used binary operators?

I was proposing moving this question to the philosophy exchange (assuming that they're open to it) and rewriting the body of the question to be narrowly focused on a single connective for concreteness. You could focus on xor, for example. You can also explain that your broader question is why some definable connectives are easy to use at a meta-level and some aren't.

@dxiv The thought of this question in my head appeared first with only xor. I don't think there is anything particularly special about xor.

@Okoyos "don't think there is anything particularly special about xor" $\;-\;$ Precisely. But then it's a bit like asking why does the positive root of $x^2-x-1=0$ have a recognizable name (golden ratio) while the roots of $x^2 -3 x - 1=0$ don't have a commonly used name.

@dxiv I think the question "Why does the golden ratio come up more often in mathematics than than the positive roots to the equation $x^2-3x-1=0$?" is a perfectly fine question.

@dxiv Actually, the positive root of $x^2-3x-1$ does have a name: the bronze ratio.

@Sandejo Thanks for that, I learned something new today. That said, "bronze ratio" must be a lot less known than "xor" ;-)

Just say either or. In general, don't overdo it with the formal notations. "Every integer is either even or odd" carries the same meaning as $\forall n\in\mathbb Z((2\mid n)\operatorname{xor}\neg(2\mid n))$. But which one is easier to parse? It's my opinion that natural language is often better, or at least just as good as formal notation. Only use formalities when they actually aid understanding.

Why don't we use xor more often in ordinary mathematics? Do you mean the specific name 'xor' or the logical connective that is sometimes named by 'xor'?

@Vercassivelaunos I would agree with your example, but I don't think it's necessary to only use xor in a formal way; instead, you could use it like in my examples: every integer is even xor odd. I should also add that "either...or" does not mean xor but rather is more often than not the same as "or"; instead, "either...or...but not both" would be the same as xor, which is more verbose.

Then maybe it's a language thing. The phrase "either ... or" translates to in my language is very specifically xor. If it's less clear in English, then some way to express it would be fine, I think.