I couldn't get the cartidge out of this old faucet for nothing. Finally, I used a torch, and it loosened up right away. This white stuff inside is as hard as a rock? It can't be tape.
Maybe soak a cloth in white vinegar and shove it in the valve. If it foams, the white is lime. Leaving the cloth for a while may (hopefully) get the lime loose enough to brush off with a nylon brush.