After the West falls, The Romans start instituting a system like the Janissaries - when they reconquer territory they take the some of children of non-Greeks (perhaps Zeno, perhaps Nepos could do this), but are allowed to retire and have children.
The Janissary system compliments the Legions and themes and creates a larger army for the Romans, whilst reducing the non-Greek population, whilst the Greek-speaking descendents of the Janissaries settle the lands emptied by the locals.
The system can be tweaked, where the choice is that a member of the family joins the theme system (typically the father, perhaps a proxy), or they enter a lottery for their child to be taken, whilst the father will be Greek-speaking, their families will likely not be, but with resettlement the families might find the Greek Lingua-Franca quite a valuable tongue to learn, and the father teaches his children when he comes back from war.
The larger manpower pool created by both turning Slav into Greek, and any captive children into Greeks helps strengthen the Empire and makes it persist, and creating the incentive to expand northwards because the East is lost (for now), say an alt-Seljuk power is strong, no point invading the strong when you can expand in the weaker areas. Dacia and Pannonia are invaded in a number of campaigns, and forcibly settled by Greeks (probably Janissary retirees, and thematic troops), with the original population scattered throughout the Empire each time.
The more European Byzantine Empire may collapse at some point, maybe it doesn't - but they've carved the entirety of the Danube valley after an institutional system of kidnapping, brainwashing, re-educating and culturally assimilating children and fathers, and redistributing the captives families.
You know, what the Romans used to always do. Plus, you have the bonus that Anatolia might speak Greek as well.