Separate autoplay from a playback failure
Apple documents per-website choices for allowing autoplay, blocking media with sound or blocking autoplay entirely. A video waiting for a deliberate Play action can be expected behaviour under those settings.
Muting a tab is another separate control. If picture plays but there is no sound, inspect the tab and output destination before changing the media account. A blocked autoplay and a muted active video are different observations.
Use website settings for the actual site
Safari’s website settings allow site-specific choices. Read the current Apple instructions and inspect the setting attached to the genuine service page. There is no need to weaken settings for every unrelated website to diagnose one playback task.
A browser service can also set its own support requirements, DRM conditions and account/session rules. Safari being installed on a Mac does not prove that every service offers a supported web player.
Keep AirPlay and local playback separate
An AirPlay-capable destination can be another output path, but it is not the same as the video playing in the Mac’s browser window. Verify the actual source, receiver and application behaviour.
If local playback is normal and only the wireless destination fails, preserve that distinction. The next useful check concerns the output path rather than reinstalling a native media application that is not being used.
Write the failure in stages
State whether the page loads, whether pressing Play starts it, whether sound is muted and whether an external output is selected. Add macOS/Safari versions. These details make a support conversation concrete without revealing passwords or a private playlist address.