Proving Undecidability with reductions - Why do some proofs not use an Oracle?
An oracle call is stronger than emulating a TM: it allows you in constant time to solve the task of checking if some $x$ is in the language specified by the oracle.
What you saw didn't use oracle machines, since it proved by assuming towards contradiction: It assumed there is a machine $M$, and showed how to use its code in order to build a new machine $M'$ solving a problem that can't be solved - hence $M$ doesn't exist.