Prove NLOGSPACE$\subset$PSPACE
What would you think about a proof that says the following?

$NP \subset DTIME(2^n)$
let $m = \log n$, then $DTIME(2^n) = DTIME(m) \subset P$
Therefore $P = NP$

Here, you have indeed $DPSACE(log^2n)\subset PSPACE$ but the argument is weird.
Also please note that Savitch's theorem already proves that $NPSPACE \subset PSPACE$, and you have the result you want, since $NLOGSPACE \subset NPSPACE$.