Cheap Chromebook fine?

I think so.

The biggest issues I've run into are the power supplies, and galvanic isolation.  Keep the laptop/PC supplies away from your power conditioned audio gear. Laptop wall warts/power adaptors are terrible sources of noise.  If you can do that and don't have humm from the USB adaptor you'll be fine.

I think the Chromebook will work fine. I use a Raspberry Pi 4 as a streamer and it's not as powerful as a Chromebook. Looks like you can install Kodi from Google Play, so you'll be able to stream using UPnP as well as play files from a music library using that app.

I streamed music from a laptop (not a Chromebook) for several years. Biggest objection was very loud fan noise from the computer. I think this would be worth researching before selecting a device.

Will the quality of the PC make an appreciable difference for me?

A more powerful computer won't improve sound quality, but it will allow computer to "get out of tthe way." so to speak. 

that means 

- no computer fan noise (silent PC or laptop) 

- a custom power plan which allows the mainboard to perform without much power consumption

- fast storage (like solid-state drives / NVMe, fast RAM, and a decent CPU for accessing large (or any) audio files quickly on playback software

Always get a good USB cable or a custom one to avoid issues with audio latency/packet errors or ghosting.