MSI- Radeon RX 580 8 GB armor video card can run most games
paired with a ryzen 7 and b450 tomahawk motherboard and 16 gb ram?
I wouldn't bother moving from a 1060 3GB to an RX 580 8GB, as the performance difference would be too small in most titles to really be all that noticeable, especially if you are running games at reduced settings. The additional VRAM can help in some titles, and will likely keep the card relevent longer for new releases, but at 1080p with lowered settings an RX 580 will only average around 10% more performance in most existing games, so long as its not being held back by your CPU. That's not going to be noticeable enough to justify an upgrade.
To be a worthwhile upgrade from a 1060 3GB, you would likely want to look at a card that offers more performance than that. The newer GTX 1660 6GB should offer over 30% more performance than a 1060 3GB when not being limited by the CPU, and would probably be the minimum graphics upgrade that I would consider worthwhile.
Of course, in demanding games like Battlefield V, your CPU is what's holding back performance. Most games can still get along fairly well on a quad-core i5, but some, like Battlefield V, perform best on a processor with more available threads. Something like an i7-7700 or 7700K could likely plug into your motherboard, and in addition to offering higher clocks, their Hyperthreading would help performance in games that utilize more than 4 threads, but they are overpriced for what they have to offer at this point. At well over $300 new, or even around $250-$300 used, they are priced so high that you are probably better off replacing the motherboard so you can go with something else. There are a number of 6-core processors priced under $200 now, like the 6-core/12-thread Ryzen 2600 and 2600X, or the 6-core/6-thread i5-9400F. The Ryzen 3600 will also be launching for $200 in a couple weeks, and the 3000-series is expected to bring around 15% more performance per clock compared to existing Ryzen processors, so you might want to wait a bit and see what those have to offer.