But there'd be a finite number of ways to transcribe a piece. This someone would be able to claim the only ways to faithfully play the original public domain piece.

In that case, the law is not incompatible with you both owning your own transcriptions, even if they are identical. You might need some proof that you created your own if sued, but that's the copyright system.

Why not copy and say it's independently created? It does make it incompatible.

Same difference. It makes 0 sense that they own anything played based on the transcription for 160 years.

Irrelevant. All playbacks of the piece will be copyright-struck, regardless of the transcription used.