Is my reasoning wrong that $PSPACE$ should not equal $EXPTIME$?
I did not understand the first part of the question, but just so you know, its unknown whether $EXPTIME=EXPSPACE$ or not. What is known is that $EXPSPACE\subseteq \bigcup _cDTIME(2^{2^{n^c}})$ from the relations shown here.
The second part of the question: If $EXPTIME=EXPSPACE$ then $PSPACE\neq EXPTIME$, is absolutely correct: by the space hierarchy theorem, $PSPACE\subsetneq EXPSPACE$ and therefore $PSPACE\subsetneq EXPTIME$ as $EXPSPACE=EXPTIME$ (by our assumption).