Reviewing blog posts revealed the source of yesterday's noise. It was the 1000uF cap which allowed the STM32 to drive the phones. The new cap required its own separate ground path to eliminate the noise. Sharing its ground with the STM32 ground retained the noise, even though the STM32 already had smaller caps sharing its ground. It was a most perplexing phenomenon.
The best idea is the 3.3V line is a very noisy power rail directly from the PI. The cap short circuits high frequencies to ground, so it could raise any shared ground. All other ground paths from the STM32 would be raised, if the cap shared just 1 STM32 ground path. The smaller caps also raised the ground, but not as much, because they couldn't transfer as much current.
Things would be a lot easier with a flood filled ground. The underside is basically a point to point board, to get the grounding to work. The extra ground path lowered the noise to dead bug territory, but also reintroduced the 2 minute spikes. The 2 minute spikes were much subdued. The STM32 ground eventually shared a mere 2cm ground wire with the analog side. Cutting this ground path from the STM32 ended the 2 minute spikes. Fortunately, the AK4524 was designed well enough to isolate its digital side from its analog side.
Single ended analog audio is pretty soul crushing, but it's actually almost quiet enough to hear aliens. Attaching the phones with 0 gain didn't increase the noise.