Two cards side by side said nothing about whether the rooms were playing
together; now a room that joins the AVR moves into its card, under the
input it is playing, and leaving puts the card back in its own slot. The
button changes with it: Join Home Cinema when it stands alone, Leave when
it is in the group, since the card's position already says which it is.
Also fixes Separate everything, which sent a GET to a POST-only route and
so failed with a 405 and snapped the rooms straight back.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>