I have a Steam Link and 2 controllers. It's a pretty convenient way to get content onto the TV, but you still have to have a PC that is up to the task of running the game, so a Steam Link by itself doesn't seem like a good solution for you. Also, it's not necessarily something you want to do over WiFi, so keep that in mind. I've got the Steam Link box connected via Ethernet, but my laptop is still on WiFi, and although it works OK most of the time, I've definitely seen some hiccups that probably wouldn't be present if I they were both using a wired connection.
As for a PC to run this, either directly connected to the TV, or via a Steam Link, I would absolutely recommend checking out PC Part Picker if you are comfortable building the PC from scratch. I skimmed the site just now, and this looks like a pretty good starting point:
Entry Level Gaming Build
I wouldn't necessarily build this as-is, but it provides a decent baseline where you can start swapping out pieces to customize it for your intended purpose. For example, I'd probably opt for an SSD. A 500GiB Samsung can be had for ~$150, which would only $60 if you're willing to live with those space limitations. If you can afford it, using an SSD as your primary drive with a secondary hard drive can be the best of both worlds.
It has a lot of interesting ideas going for it:
- The shape is somewhat inverted compared to Microsoft/Sony/Nintendo's traditional controllers.
- There's no right stick. It uses a capacitive touch pad instead, similar to a laptop.
- This works as both a right stick alternative and a mouse alternative, but it isn't great at either.
- It uses a pair of paddles instead of shoulder buttons (it still has triggers.)
It's fine, but I'd probably prefer my Xbox One Elite Controller, or a standard Xbox One controller over it.
It's worth noting that the newer model Xbox One controllers are Bluetooth capable. I haven't verified it yet, but I have read that it is possible to pair them with the Steam Link, and it should be possible to do the same with a Windows or Linux box.
You can get an Xbox One S for ~$200 without too much trouble. It can't do 4K gaming, and 60FPS is going to depend on the game, but it's a fantastic entry point if you aren't already tied into another ecosystem.
It's called Steam Big Picture Mode.