You basically just described administrative government. Which brings up to THE question; how to differentiate Republics from Administrative?
Perhaps the Republic could force everyone to have their estate in the capital, with estate owners called Patricians.
The Doge/Stadtholder would be more limited compared to the Basileus. Maybe all Patricians get to vote on stuff the Doge does (imprison, declare war, etc) in a Senate.
And Patricians would be able to build special buildings in any province where the local liege accepts, things like banks or warehouses, which open unique interactions. That's where they get their money to influence the Doge.
Republics would also be very restricted regarding expansion compared to Administrative. If the latter is for wide play, the former would be for tall play.
Administrative government is more based on governorships of the provinces. For a Republic, it needs to be based much less on governorships but rather the main republican offices and participate in the assembly/curia. So having the different College of the Sages, Captain General of the Sea and so forth. Holding offices should come with actual powers and not just giving some buffs to the republic.
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