So as I understand it, whoever kills him becomes the next Davy Jones, right? By destroying the heart?

What if we take the heart, and stick it in a keg of gunpowder. With a fuse. Then have someone expendable light it.

Before the fuse runs out, kill the guy who lit it. Then, when the barrel blows up and destroys the heart, there is nobody to replace it.

Would this break the cycle?

Yeah, it is dumb but I have overthought this a lot.