steam link is probably a better fit for you then

My main workstation is powerful enough to play Steam games, and a Steam Link would allow me to stream them to my living room TV. Where that fails is when I have work to do on my desktop and my kids want to play games.

To solve that, I had to build a dedicated living room PC. I didn't go nuts, so it ended up being about $500.

If I could replace that with an handsome Linux powered gaming system for less than $300, then I think it's a no-brainer... especially if it can run third-party marketplaces and apps like Steam or Plex.

What does the living room PC get used for?

If just games/movies/music then what was the logic for getting that over a console? (PS4, XBox1, Switch are around the same price point or below) Just curious, not suggesting that a console would have been better.

Why would you build a living room PC instead of building each child their own Personal Computer? Then they could hook it up to the Link if they wanted as it is their PC.

Because the child's personal computer resides in their room. This means less room (no pun intended) for control over active times, more costs involved because of computers not being shared, and also enforces the child to be excluded from social environment. It is a double edged sword. Both options have their pros and cons, and none is ideal.

Why would it be in their room. We have all our computers in one room, our family office. A shared, social environment.

You're right.

This seems like a hard problem, esp. since OP is so emphatic that $300 is preferrable to $500. Let me see if I can figgure this out for us both.

1 child x $500 = $500

1-or-more x $500 = $500-or-more

Oh! That must be it! This solution would be more expensive to implement, and therefore, since cost is a factor, this solution would be less preferable!

Glad I took the time to work this one out, I certainly was very confused as well.

Money isn't always the primary factor.

I bought Steam Link. To my slight surprise the hardware worked fine and graphics were smooth (in wired LAN).

Where it fell short was the overall experience. Most games are of course not designed for this. In PC you always have the keyboard, there are things that are tweaked through Windows. Also it felt that I had to regularly run to the PC to sort out some issue. Disclaimer: I tested just with few games and at least 6 months ago.