Refactor and extract components out of Sync.razor

Sync.razor has become bloated and unnecessarily large. Having things separated out into a YouTube player component is going to make supporting multiple video platforms/formats easier. Should also rename sync.razor to room.razor in the process to clarify what it actually does.

Start by reading Sync.razor and tracing the responsibilities it currently contains. Separate the YouTube player into its own component and rename Sync.razor to Room.razor, preserving the existing behavior. Done means the room view remains functional and the extracted structure supports adding other video platforms or formats.

Written by the indexing model from the issue text.