Linux

I guess you might want to ask this question in one of the self-hosting communities (there's a bunch that are active, so I'm not linking one)

But getting back to the topic:

• jellyfin needs very little
  I'm running mine on a laptop with 4 threads at 2.6GHz and on 8GB of RAM it was fine with a bunch of other stuff running
• OwnCast
  IDK but for personal use I'd expect to also have low requirements
• Minecraft server
  That part I'm not sure about. I'd expect it to be a bit more demanding
• play some steam games
  I understand you'll be running those on another computer and only streaming via OwnCast, so that goes back to earlier point

Bigscreen only just came back around, I wouldn't go too heavy there yet (and I'm testing it out already myself)

Get flex launcher instead. Simple config, does the job.

In terms of specs, there is a lot in your list that can vary, and make those specs jump. Where to start, IMO, would be a 7th gen Intel. If an 11th gen is an option, there and up transcoded av1.

For storage you're going to have to do the math yourself in terms of what you'll have. I have 40tb of media. I used to have a 1tb drive for everything. Storage is how much you want to allocate, nothing more or less.

Ram is similar. Find the worst case game, most overloaded with plugins Minecraft server or whatever, and start there on memory use. JF won't take much, your game choice might.

Plasma Bigscreen is kind of a mess still, so maybe run it on something first to understand what you're getting into. That being said, any distro that runs KDE can run KBS, but will probably not be tuned for the least amount of resources to be used.

Something like OpenELEC/LibreELEC or OSMC are very well tuned for this, and require very little in the way of resources though. I'm not saying DON'T use KBS, but if you're lacking in resources, the others are probably a better choice.