Q:

Why are function definitions not written with the := sign

Why are function definitions not written with the $:=$ sign instead of the $=$ sign.
It seems to me that $:=$ would be more intuitive and avoid a lot of unnecessary ambiguity.
Consider the following example:
$$f(x,y,z) = x^2 + y^2 + z^2$$
$$f(x,y,z) = 0$$

The first line - defines a real valued function who's output is the sum of squares of all of its parameters $x,y,z$
The second line - either also defines a real valued function who's output is always $0$, or more commonly, is asking us to find the roots of some function $f(x,y,z)$

Notice the subtle difference between the meaning of the 2 equal signs. So, would it not be more appropriate to instead write the first line (the actual function definition and not root finding) with the $:=$ sign? (since we are defining it, after all)
$$f(x,y,z) := x^2 + y^2 + z^2$$
$$f(x,y,z) = 0$$
Is there a reason why we don't do this? Am I totally mistaken on some subtle detail in the notation?

A:

First of all it is totally fine if you want to use $:=$, and there certainly are people who systematically do it.
What you have to cosider is that both your formuals do not make much sense without any context. Example:
$$f(x,y,z) := x^2+y^2+z^2$$
It looks like $f$ is supposed to be a function, but what set is it defined on? This is crucial. If we know that our subject "takes place" in $\mathbb R^n$ it might be unnecessary to specify this, but it might also be just a substet who knows?
$$f(x,y,z)=0$$
This is very unclear now without any contex, is $f$ just the zero function? Are $x,y,z$ defined?
With the corresponding context it is usually quit clear what your equality wants to express, e.g. equality as a definition, equality as two equal values e.t.c.

EDIT: To the comment:
The first thing to notice: You can consider all textbooks and papers as written as a text that you can continuously read through. So here your requested examples:
Defining a function
(Context: Real analysis)
Choose any open set $U \subset \mathbb R^3$ and define $f: U \to \mathbb R$ as $f(x,y,z) = x^2+y^2+z^2$. It is obvious that $f(x,y,z) \geq 0$ $\ldots$
Alternatively if the domain does not really matter or is obvious you could just write: *Let $f(x,y,z) = x^2+y^2+z^2$.
E.g. in this textbook.
Specifying equality
An equality is usually the implication of something that came before so it should be qute clear what we're talking about.
(Assuming we have defined $f,x,y,z$ before) $\ldots$ Then $f(x,y,z)=1$ holds.
Finding roots
As said above, if you just write $f(x,y,z)=0$ nobody will understand what you want to say. As always it should be written as a sentence.
Or if we want to look at the roots: *$\ldots$ Then the roots of $f$ are given by $\ldots$
Or
$f(x,y,z)=0 \iff x^2+y^2+z^2 = 0 \iff (x,y,z)=0$

