Your argument, which can be rephrased as:
There are only countably many real numbers which could ever be defined, so all the rest are useless
is fallacious. There are plenty of times in mathematics when we want to deal with large sets, without caring whether we can explicitly define all of them.
As an example: compact Hausdorff spaces have properties which make them interesting to mathematicians. But any countable example must have an isolated point, and we'd like to know if there are examples without isolated points.
An obvious example of a compact Hausdorff space with no isolated points is given by closed interval of real numbers, such as $[0,1]$. We can't define every element of $[0,1]$, but we can still say interesting things about it - for example, it is compact.
On the other hand, let $A$ be the set of all definable elements of $[0,1]$. Then $A$ and $[0,1]$ have precisely the same sets of definable elements, but $A$ is not compact (by the Baire category theorem). So even though we can't define any elements of $[0,1]\setminus A$ explicitly, these undefinable numbers do have an effect which we can define.
Why is this? One answer comes from talking about induction principles. You're probably familiar with the principle of mathematical induction - if $T$ is some set of natural numbers such that $1\in T$ and $n\in T\Rightarrow n+1\in T$, then $T=\mathbb N$ - but there are versions which apply to much larger sets than the natural numbers, and even to objects like the class of all sets which are too large to be sets. The $\in$-induction scheme is the following:
Let $\phi$ be a property of sets such that whenever $\phi$ holds for all elements of a set $x$, it must hold for $x$ as well. Then $\phi$ holds for all sets.
If you're used to thinking about induction in the following way:
We have $1\in T$, so we must have $2\in T$, and then we must have $3\in T$, and so on...
then it might seem confusing to induct over all sets. There are certainly examples of sets that can't be defined; indeed, any real number has a unique representation as a set. Yet we can still induct over all sets; it seems as if mathematics is somehow allowing us to make infinitely many calculations at once.
The reason we can do this is that induction principles are proved using arguments by contradiction, which turns out to be an extremely powerful tool. I believe that in constructivist logic (where statements can be neither true nor false) then there really is no extra benefit that you can get by including numbers that can't be defined, so your question does get to the heart of some pretty deep mathematical logic.
For further reading, have a look at this fictional dialogue written by Timothy Gowers.