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Here’s a similar question, my answer should also be relevant to your question.
As for Switch emulation, yuzu still runs great, with many playable games on the deck. If you don’t have the AppImage or Flatpak already, make sure to get it from a reputable source.
Edit: Or, if you’re lazy, just use Ryujinx (also available in EmuDeck).
You could try getting a Raspberry Pi Zero together with some kind of SPDIF output card, but that will probably go over $30.
I have no idea what pricing is like, but you could possibly try getting a used Logitech Squeezebox player.
If you’re desperate to stay on the cheap and don’t mind BT quality, you could also install Snapcast on an old phone, enable the Snapcast player provider and then use the phone to connect to your speakers over Bluetooth.