Having a single, central receiver to collect inputs from 5+ A/V sources and deal them back, arbitrarily? to 6 different outputs seems awfully complicated. Something like Jellyfin or mythtv with one or more TV tuners would let you originate all of the TV & streaming signals from your central source to client players - kodi or whatever - in the player rooms. Most of those have at least some control through homeassistant. Kodi on RPis with some basic class D amplifiers in each room, run through the TV, if the room has a TV. Probably couldn’t get synchronized audio in all rooms that way.
My first server was a single-core Pentium - maybe even 486 - desktop I got from university surplus. That started a train of upgrading my server to the old desktop every 5-or-so years, which meant the server was typically 5-10 years old. The last system was pretty power-hungry, though, so the latest upgrade was an N100/16 GB/120 GB system SSD.
I have hopes that the N100 will last 10 years, but I’m at the point where it wouldn’t be awful to add a low-cost, low-power computer to my tech upgrade cycle. Old hardware is definitely a great way to start a self-hosting journey.