why this is a problem? Streaming over Http just works - even on clients from 1999 like Winamp.

Why would you military grade encrypt radio service or static site anyway?

To keep your ISP from slipping ads into the audio stream? To keep RIAA from finding out what you're streaming and suing you into oblivion?

And, more importantly, to keep misconfigured network appliances from treating every resource on the web as HTML and trying to shove ads into it, breaking audio players.

Because some browsers get annoyed and put up lots of alerts if they can't make a https connection

Rusty, of SomaFM fame? What a blessing to see you here! Mega fan, proud SomaFM mug owner from Australia :)

A static site? Because any hop can inject arbitrary code which your browser will then execute.

TLS doesn’t just offer confidentiality, it also offers integrity.

It doesn’t “just” work, I just implemented a streaming radio client and the http streams were a nightmare

Running my own icecast2 server, I recently ran into a problem[1] where TLS connections don't shut down properly. It's actually a problem with the libshout client library's poor TLS support. I posted a patch to that issue, but it's hard to tell if there's anyone still looking at issues or actively developing the project over there. The last update to libshout was 3 years ago, and it was just a documentation cleanup.

1: https://gitlab.xiph.org/xiph/icecast-libshout/-/issues/2337

This should be working in tinyice without issues. Feel free to test it and if the problem is also present, I will fix it.