If $PSPACE^{SAT}=PSPACE$ and $PSPACE \subseteq EXP$, then why does $EXP^{SAT}$ not necessarily equal to $EXP$?
The fact that the base class can simulate the oracle does not mean that the oracle does not give additional power. Consider the polynomial hierarchy, $\mathsf{NP,NP^{NP},NP^{NP^{NP}}},...$ at the $i$'th level you get an oracle for the $i-1$ level, so following your line of thought we should not gain additional power. However a naive simulation of the oracle does not suffice, since taking the wrong path may lead us to accept an input outside the language (we need to be sure of the oracle's answers).
A class which is famously not low on itself is $\mathsf{EXP}$, since $\mathsf{EXP^{EXP}=2EXP}$. A naive simulation in your case fails for similar reasons, as a machine in $\mathsf{EXP^{SAT}}$ can ask the about the satisfiability of exponentially long formulas.