You'd probably have someone adopt the philosophy as the forming idea of the state. People aren't a fan of having their families dissolved.
I'd put in a Philip Lives timeline, if the PoD is that Philip agrees rather than disagrees with Plato. (IOTL they distinctly disagreed on this topic.)
Have him take captives, and use them to create a caste of people completely outside the family structure. Raised by the King and his government, under that principle. More or less to break any resistance to his rule in his new Empire. The key is to take them as children, and through them organise them in the Gold/Silver/Bronze hierarchy. Furthermore, once established, you can always offer families the ability to give their children over for some money - particularly the poor, who in turn expand this system.
Eventually that caste can overtake the rest of society, first by forming a large loyal force for the King, later expanding to be a polity within a polity, which then supercedes and tears apart the remnants of the parent polity.
But that slow takeover, over probably over generations, is the key. Break family bonds. Reorganise. Overwhelm.
Then in theory, assuming you have a series of rules for establishing a King (heck, the inheritance might be how this society fully forms, perhaps even as a breakaway), it can continue.
Personally, I like the idea of Philip trying it out as a way to replace Byzantion, since the site is so good - and turning that location into the heart of the system - taking 1000s of captive children, bringing wealth, bringing teachers of all kinds, and settling them there, and using The City.
The lingering problem is how you deal with the established Elites - and all I can think of is granting them all "Gold" status, and after a point, if they try and keep their children, crushing them ruthlessly. That will include the royal family however, so unless you have some sort of adoption by Alexander (perhaps as a way to supersede his other rivals IOTL as heir), then I'm not sure how it survives Philip, unless the Royal Family is outside the system until it reaches critical mass. (Perhaps a Bronze-Quality King reveals themselves).
But yeah, Alt-Philip choosing to try the idea to integrate his new lands COULD be a PoD to try this out.