So from what I have gathered it is a fairly standard result that if $f: X \longrightarrow Y$ is a finite morphism of schemes, then it is quasi-finite. By quasi-finite here I simply mean that the preimage $f^{-1}(y)$ is a finite set.
The standard method for doing this, it seems, is to first reduce to the affine case, which we can do for a finite morphism, so that we have $$ f^{-1}(\text{Spec } B) = \text{Spec }A $$ via a morphism of rings $$ \phi: B \longrightarrow A $$ so that $A$ is finitely generated as a module over $B$. One then argues that the prime ideals lying over a particular $\mathfrak{p} \subseteq B$ (corresponding to the point $y$) are in one to one correspondence with the prime ideals of $$ \kappa(y) \otimes_{B} A, $$ where $\kappa(y)$ is the residue field of $y$. This can be shown to be a finitely generated module over $\kappa(y)$, or in other words, a finite dimension $\kappa(y)$-vector space. The proof then concludes via an argument that vector spaces are Artinian and that they have finitely many primes.
What I don't understand is why such an argument is used when it seems to involve a fair bit of background machinery, and only says that the preimage is finite.
Why can't we simply say that since $\kappa(y) \otimes_{B} A$ is finite over $\kappa(y)$, then it is integral over $\kappa(y)$. Then any ring integral over a field is itself a field. But then $\kappa(y) \otimes_{A} B$ has exactly one prime ideal. But no reference seems to use this argument, and most seem to strongly suggest that in general there is more than one prime ideal. So where exactly does this argument fail?
See Integral extensions of rings, when one of the rings is a field.
It's not true that any ring integral over a field is itself a field. What is true is that any integral domain integral over a field is itself a field. But $\kappa(y) \otimes_B A$ may be finite and integral over $\kappa(y)$ without being an integral domain.
For example, $\kappa[x]/(x^2)$ and $\kappa[x]/(x^2-x)$ are finite and integral over $\kappa = \kappa(y)$ but not integral domains. The first one has exactly one prime ideal despite not being a field, but the second one has two prime ideals.