It sounds like you are currently using integrated graphics, so it's probably worth upgrading soon if you are interested in gaming, rather than sitting around with a system that isn't quite up to the task. Intel's current iGPUs are bottom-of-the-barrel as far as gaming performance is concerned. Your CPU is pretty good, but without a graphics card your performance in most recent games will be abysmal. Something like a GTX 1660 should offer somewhere in the vicinity of 10 times the performance of Intel's UHD 630 graphics, and should handle all current games rather well at 1080p60.
AMD will be launching a new generation of cards soon, but so far they have only announced models priced upward of $380, which are set to come out next week. It's unclear when more mid-range models will launch though, and it could potentially be months before we see any new cards from them in the $200 to $300 range. Considering the pricing and expected performance of their announced cards, I have doubts that they will be pushing performance-per-dollar all that much beyond what we see from existing models.
As for the RX 580 and 570, they are also decent options that offer good value, though they have greater power demands than a GTX 1660, while offering less performance. They are still good options at lower prices though. In terms of how their average performance compares across recent games, an RX 580 is around 15% faster than an RX 570, a GTX 1660 is around 15% faster than an RX 580, and a GTX 1660 Ti is around 15% faster than a GTX 1660. These numbers can vary a fair amount from one game to the next though.
The RX 570 technically offers the best performance-per-dollar of the bunch, but it falls a bit short of allowing many recent games to run at full graphics settings at 1080p while maintaining 60fps, and future games are bound to get even more demanding, so I would personally want something a bit faster. It's arguably a very good value option considering the low price though.
With a budget of around $200, I would currently pay the bit extra to go with a 1660 over an RX 580. In addition to being faster, it's also more energy efficient under load, so it should generate less heat, which could potentially make it quieter to cool (depending on the card) and less demanding on the system's power supply. Plus, if the system were used for gaming for a couple hours a day on average, that could add up to over $20 in electricity over the course of 3 years (at the average cost of electricity in the US), effectively bringing the total cost of the two cards closer together.
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