Yes, I think you are right (in fact, it would be interesting, since the parallels between this and his own life would be clear - the legitimate king is forced to abandon his throne in favour of an upstart military commander and seeks protection to the court of the ruler of the other great power of the East). What I am saying is that perhaps Phokas manages to neutralise the heirs and thus prevent them from trying to escape to Persia, in order to make it more possible to avoid a war erupting right after Maurice's death.
Also, again, I agree that Khosrau most likely wanted war - he must have viewed the concessions he made to Maurice (which basically undid almost 30 years of policy and wars) rather excessive later on. But without Maurice or the heirs, he might have waited for some time and struck one or two years after OTL, which could give the empire some more time to bolster its defences (if Phokas scales back the operations against the Avars, he might move part of the forces Maurice had transferred to Europe after the peace with the Persians back home, meaning that the Eastern armies would be in a somewhat better shape).
Yeah but why would he wait?