I’m concerned about a particular part of the proof: Why do most proofs of this theorem involve the canonical homomorphism. What does that add to the proof? (Nothing from what I can see.) What’s wrong in just showing that there is a well-defined isomorphism, namely $\Phi : R/ker\phi \rightarrow im\phi$ defined by $[r]\mapsto \phi(r)$.
I know how to prove this theorem. I’m just interested in the purpose of people introducting the canonical homomorphism.
One reason to explicitly use the canonical homomorphisms in the various isomorphism theorems is that we can then interpret the results using the language of commutative diagrams.
Personally, this abstraction first became important in homological algebra. I recall at least one proof which was completely unwieldy if you tried to follow how to define a map element-wise, but became almost trivial if you rewrote the map as a composition of natural maps coming from commutative diagrams (e.g. snake lemma, five lemma, or more generally limits/colimit diagrams, etc.).