Safari Private Mode causes Mac to sleep while video is actively playing

You've correctly identified the root cause: Safari Private Mode intentionally suppresses certain background APIs, including the power assertion that tells macOS "media is actively playing, don't sleep." This is a known behavior in Safari where Private Browsing mode has stricter restrictions on what web content can do in the background, which inadvertently prevents the sleep-prevention signal from being sent.

This is a confirmed bug in Safari. Apple has been aware of this class of issue since it appears in multiple macOS versions. Your workaround (enabling "Prevent automatic sleeping on power adapter when the display is off" in Battery settings) is currently the most reliable fix.

Additional workarounds:

1. Increase your display sleep timer:

- System Settings > Displays > Advanced (or Energy Saver > Prevent automatic sleeping when the display is off)

- Set "Turn off display after" to a longer time, or "Never" when on power adapter

2. Use a third-party app to prevent sleep:

- Apps like Lungo, Amphetamine, or Caffeine (all free/cheap on Mac App Store) can keep your Mac awake on demand without changing system-wide settings

- This is more flexible than changing system settings

3. Use a non-Private tab:

- If privacy is the concern, consider using a regular tab after clearing cookies, or use Firefox's Private Mode which handles this differently

4. Report the bug:

- Please report this at feedbackassistant.apple.com under Safari > "Safari Private Mode does not prevent display sleep during video playback"

- Include your macOS version and Safari version

This behavior has been confirmed on multiple versions of macOS/Safari and the Feedback reports help Apple prioritize the fix.