Is my reasoning wrong that $PSPACE$ should not equal $EXPTIME$? It's impossible for a problem to require exponential space without being exponential-time.

Consider that if an $EXPSPACE~~complete$ problem can be solved in $2^n$ time. It will now fall into the class $EXPTIME$.
Then $EXPSPACE~~complete$ problems are in $EXP$ if they can be solved in $2^n$ time. This means they can reduce into $EXP~~complete$ problems and vice versa.

To me, this should be easy to write a proof that $EXPTIME$  = $EXPSPACE$.
My intuition tells me that if $Exptime$ = $Expspace$; then $PSPACE$ != $EXPTIME$,
Because $PSPACE$ already is not equal to $EXPSPACE$.
Question
As an amateur, what would make this reasoning be wrong or right?
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).