Ignoring the possible subsequent bedroom use I would suggest to try to spend as little as possible on electronics (let alone cables) and devote as much as possible to the speakers because they are what produces the difference in sound quality.
The Chromecast Audio is the cheapest and most versatile streaming option and the analogue output is more than good enough for a system at this level. The Sony receiver that I mentioned is similarly good enough at this level, and a real bargain. There are similar amplifiers from the other major Japanese brands and these represent the best value for money. Speakers are a matter of taste - none will be perfect. Here, the US market is quite different from Europe with different brands and different relative prices, so it is hard for me to make a specific recommendation. Over here, I would have said the Wharfedale Diamond series is always very good value for speakers that have few weaknesses. I particularly like the Wharfedale 220, but there may be good alternatives in the US, I don’t know. A few years ago I bought some Tannoy DC6 standmounts for my daughter, and they sounded even beter (at a higher price). But it is a volatile market.
If you audiotion, make sure to listen at equalized level: the human brain will interpret the loudest speaker as the best speaker.
Another very simple and compact option would be the recently discontinued Quad 9AS active speakers. If you can still find a pair at a discount price, they would be worth it. They have three digital inputs, so if you add a Chromecast Audio via its optical digital output you are all set for maybe half your planned budget. Use the coax input for your tv and you will also have far better tv sound. See here, e.g.:
https://www.musicdirect.com/speakers/quad-9as-active-desktop-speakers