I found that Dolphin uses OS sleeping to run the emulation roughly in sync with wall-clock time, and use a dynamic-rate audio resampler to output sound with a bit of added latency. This has the complication that frame delivery is no longer synchronized with monitor refresh, and requires VRR displays (unlike an actual Wii) for low-latency output. I had a draft PR to loosely couple the sleeps with "frame queuing delay", but it didn't get merged.

In the case of writing an audio synth, I think the usual approach is to "hurry to sleep", but synchronized with the audio device/OS mixer's 2-3 audio buffer periods/pages (rather than visual frames), and asynchronously update the UI. Are you planning to use wall-clock time, or the audio sample clock, as the source of truth for timing?