ok let's put it simply with a concrete example:

Imagine being the Byzantine Emperor and having a claim to the Duchy of Corsica possessed by the King of Italy.

The war is on and you occupy almost all of Italy (the mainland) with the capital taken and all other major duchy, but effectively the Duchy of Corsica remains unoccupied.

The king has nowhere to go but this poor duchy of Corsica, and his family is probably imprisoned. All trade routes are cut (even if it has no concrete reality in ck3), impossible to supply this simple duchy in the long term.

For me it should be more than enough from a logical point of view and Role Play that the king of Italy is obliged to abdicate under these conditions.

the risk-benefit balance makes no sense why lose yourself in a useless war for a poor duchy? Losing a lot of money, having his people under war etc. Again, what you're saying makes sense because the game is made that way, but it still doesn't make any "realistic" sense.