Well, let's talk about algorithms that cannot be represented as a finite bit-string for any kind of encoding.
Let me type out such an algorithm for you... Ah, but if I do that, I can represent that algorithm with the encoding of my typed text.
How about representing my algorithm using some 'analog means', say by the position of a few coins on my desk. Although the position of those coins can be modeled by some real numbers (which could in some encodings be impossible to finitely represent), this entire description can again be considered an representation of my algorithm and can be encoded to a bit-string again!
I hope that these examples make it clear that if some algorithm cannot be represented by a finite bit-string we have no means of describing this algorithm at all!
So, why would we consider the existence of something we cannot speak of? Perhaps interesting for philosophy, but not for science. Hence, we define the notion of algorithm such that it can be represented by a bit-string, as then we at least know that we are able to talk about all algorithms.
Although the above answer the question asked, I think the confusion about the example given is mostly due to the fact that a representation only needs to uniquely represent some algorithm. The manner of representation doesn't need to involve the actual computations invoked by the algorithm! This is very useful, as it means we can also represent uncomputable algorithms!