Creating a new user didn't help. Same problem occurred under that new user account.
But I did identify the cause of the problem, and a workaround.
My Mac is connected to an early 2000s Apple 30-inch Cinema Display with a resolution of 2560 × 1600. That's more than the DVI standard supported on a single display, so Apple had to use the dual-link DVI protocol.
(If you're curious about dual-link DVI, the Wikipedia article on DVI explains it. Just skip to the last paragraph of the Technical Overview section.)
To drive this monitor, a standard USB-C-to-DVI adapter wouldn't work. I needed a dual-link DVI adapter, and I chose one sold by Club3D. There are two versions of the adapter:
- CAC-1510 (HDCP On)
- CAC-1510-A (HDCP Off)
HDCP is a copy-protection standard supported by modern monitors with HDMI and DisplayPort connectors, and some (but not all) DVI-D displays. Club3D specifically recommends the HDCP Off version of the adapter for use with Apple Cinema Displays, since some of the don't support HDCP.
Suspecting that this was an HDCP issue, I first tried just moving the browser window from my big external display to the internal display, but that didn't make a difference, even after I re-loaded the browser window.
But as soon as I disconnected my Cinema Display, and reloaded the browser window, it started working! Why the Display player enforces HDCP in Safari, but not Chrome, is a mystery to me.
Amazon Prime Video, which also didn't play in Safari on my Cinema Display, but the same fix that worked for Disney+ worked for Prime Video.
(In case you're wondering, I started playing a video on the built-in display, then plugged-in the Cinema Display to see if that would work; it didn't. Within 5 seconds of plugging in the Cinema Display, the video stopped playing and displayed an error.)
Speculation: If I plugged in a monitor that supported HDCP, I bet the videos would all play fine. I just wish anybody at Apple or Disney knew about this!
I hope this helps at least 1 other person!