I purchased a Donzi 35 ZR last season, which had previously been used in salt water. This winter, I’m in the process of upgrading both the engines and the outdrives. During the teardown, I noticed that the seacocks are seized in the open position.

What is the best way to address this issue? How is this component intended to be used, and are they necessary for my setup? I plan to run approximately 680 horsepower per engine, so I want to be sure everything is properly configured and reliable.

Depends on where these are located.....if they are on the bottom of the boats hull and feed the sea strainers or engines then the valves are there in case you rupture a feed line and start flooding the boat. You shut the valves closed to keep boat from sinking!
If they are just on the sea strainer then most likely they are on the incoming water line and they are there so you can close the valve and be able to open the strainer lid for service work or clean out while in the water...again so you don't sink the boat
Im sure they were put there for a reason so if they dont work i would replace them with new.