Captain, I think you are treating the words and tune in the same way as the recorded performance whereas I think (as a layman) that they are different.

The words and tune are in the public domain so are not owned by anyone (for instance if Sharp or Kennedy paid a royalty to the source, should that source pay their source and should that source then pay their source and so on and on). Sharp's piano arrangements were written by Sharp based on this public domain material. Sharp therefore had the ownership of that arrangement but not the ownership of the song the arrangement was based on. Similarly, any arrangement that Kennedy wrote was his property too.

To my way of thinking where the song is recorded by Kennedy on to a tape machine, for example, it is the performance that is being copied and sold on, not the words and the tune. This is owned by the source not the collector unless contracts are written, agreed and signed handing the ownership (or some other arrangement) to the collector. I think this is where a lot of the controvesy over Peter Kennedy in the other thread is really about. Whether he had the permission to do with the recordings what he did and if so did he honour these agreements.