If NP = coNP, then the Polynomial Hierarchy collapses to its first level (NP). Intuitively, it seems to me that PSPACE should collapse down to NP as well.
As a loose heuristic argument, take the PSPACE-complete problem QSAT (which is SAT where the boolean variables can be quantified with either $\forall$ or $\exists$). Since both $\forall$- and $\exists$-type branching can be handled by NP (if we assume NP = coNP), this problem should be in NP.
Yet in a historical sense, it seems unlikely to me that NP = coNP really implies P = PSPACE. If that were true, complexity theorists would never have tried to prove P $\ne$ NP, since they could instead try to show the "easier" claim P $\ne$ PSPACE, which would imply NP $\ne$ coNP, which would imply P $\ne$ NP.
So what's wrong with my thinking?