I can't seem to figure it out. When it plays a MIDI file, it references Timidity++ in the output:

  Metadata:
    name            : Piano
    instrument      : Timidity GM patches
    sample          : Piano
                    : 0:
                    : 1:
                    : 2:
                    : 3:
                    : 4:
                    : 5:
                    : 6:
                    : 7:
                    : 8:
                    : 9:
                    : 10:
    extra info      : 51 patterns, 26 channels, 1/11 instrument, 12/12 samples
  Duration: 00:03:06.00, bitrate: 1 kb/s
  Stream #0:0: Audio: pcm_s16le, 44100 Hz, 2 channels, s16, 1411 kb/s

Playing the file with Timidity++ results in nothing, as Timidity++ apparently isn't a soundfont, and searching "Timidity GM patches" brings up nothing.

No, Timidity++ is not a soundfont, it's a synthesizer:
https://en.wikipedia.org/wiki/TiMidity++

So, your installation found that Timidity++ was available and used it to synthesize the required sounds, no sound font needed (TiMidity itself can make use of available sound fonts though).

When ffplay played it Timidity wasn't installed. When I install it system-wide and use the system install I hear no sound when I tell Timidity to play it...

https://timbrechbill.com/saxguru/Timidity.php Looks like the default soundfont is TimGM6mb.sf2 . But maybe it does not come with a default, and the default is only used with the official installer. I installed Timidity++ on my Archlinux based system and I do not get an audio from it too. Archlinux recommends to configure the setting to point it to the soundfont: https://wiki.archlinux.org/title/Timidity%2B%2B

ffplay maybe uses its own configuration to load a specific soundfont. That's why you get it there. Maybe. I don't know how ffplay functions.

Maybe ffplay doesn't use timidity application. The name says "GM patches" under "instrument". Sounds to me like some Timidity based soundfont, not an application.

Tried TimGM6mb, but unfortunately that's not it. How would I find what soundfont ffplay is loading? I tried looking in the source code but found nothing of use.

I don't know. Looking up man ffplay and the webpage https://ffmpeg.org/ffplay.html and searching for the term "midi" didn't bring up anything.

Note: I can't share the specific MIDI file I was using as it's private, but in case it helps find the soundfont, here's a sample of ffplay playing onestop.mid from Windows ME. It sounds like chiptune at first, but after using the private MIDI file I can confirm that it isn't chiptune. The piano patch sounds kinda like an organ?

Here's a link to the onestop sample (Warning: it becomes ear grating after about 20 seconds due to ffplay misinterpreting percussion as notes; this is why I chose this specific song): https://files.catbox.moe/2ecdhq.mp3

This is what ffplay says when it plays onestop.mid: https://pastebin.com/SVuJai0H

The sounds are coming from patch files loaded via PAT_Load_Instruments in libmodplug, which is what's used to generate the sounds.

You are right though that the patch/instrument consists of several pre-generated samples and does not use Timidity "live".

FluidGM is a popular soundfont to bundle with distros, but in the case of Arch I'm not sure. I'll try out the package later, i might be able to tell.