Bingo! Thank you. Turning off silent mode and refreshing the page fixed it.
That's very strange though, I wonder if it uses a weird sort of web calling API or something for the audio. I've never had an issue with audio in web pages before, and even the youtube video the page embeds sounds fine. It seems quite poor UX (of Safari? Chrome?) to require turning off silent mode – many people never turn it off so it's not very discoverable, and then forgetting to turn it back on again could be pretty bad.
iOS has changed over time, such that silent mode affects notification and interaction sounds but not multimedia - I can still listen to music or watch videos regardless of whether the phone is on silent or not.
There is no other media content that is disabled by silent mode. My mental model for silent mode is that it silences things outside my control - like calls - but not things that I explicitly ask for, like music.