What is HDR, anyway?

Is there any workflow that can output HDR photos (like the real HDR kind, with metadata to tell the display to go into HDR mode) for photos shot with a mirrorless and not an iPhone?

Yes. For example, Lightroom and Camera Raw support HDR editing and export from RAW images, and Adobe published a good rundown on the feature when they introduced it.

https://blog.adobe.com/en/publish/2023/10/10/hdr-explained

Greg Benz Photography maintains a list of software here:

https://gregbenzphotography.com/hdr-display-photo-software/

I'm not sure what FOSS options there are; it's difficult to search for given that "HDR" can mean three or four different things in common usage.