Ryzen is one of the better options for gaming/streaming, mainly due to the higher core/thread count over Intel for the midrange chips. 1700x is a great chip, and games/streams like a monster with excellent in game settings with good FPS, and superb stream quality even on the highest presets.
As InvalidError pointed out, it would require a new Mobo/CPU/Ram. Yes, it does put you out of budget, but you could use the RX580, PSU (if it's decent) and case, and add this:
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD - Ryzen 5 2600 3.4 GHz 6-Core Processor ($164.99 @ Amazon)
Motherboard: ASRock - B450 Pro4 ATX AM4 Motherboard ($93.88 @ OutletPC)
Memory: Corsair - Vengeance LPX 16 GB (2 x 8 GB) DDR4-3000 Memory ($114.99 @ Amazon)
Total: $373.86
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2019-02-28 13:41 EST-0500
Although out of budget, it's not unreasonable to consider saving for a while. Along with what you have, and adding the new platform base, you'd have a pretty good 1080/60 gaming system. You would be able to stream very well at 1080p, with quality in game settings at that res, and excellent stream output and very little by way of dropped frames for the stream. It would be a great place to start.