This is an old comparison from when the Nvidia 3000 and AMD 6000 cards were first out, but this should give you an idea of what to expect from various cards that were around then.
Your 1060 could be replaced by a 6600 (non-XT) or an RTX 3060, and you'd see around a 2x increase in performance. I have a 3060ti, and it's been able to handle pretty much everything I throw at it.
You could also check into the latest Intel budget card, whatever it's called. It's supposed to be really good on Windows (but sucks on Linux, at present).
Oh yeah, arc battle mage requires resizable BAR and a few other newer cpu/pcie memory tricks to perform correctly that make it not great on cpu's older than maybe 10th/11th gen intel or ryzen 2.
Does it have to be the 9600? iirc the B580 works with 10th gen. I'd guess you're probably not buying new but the new prices of the 9600 and 10600KF are very similar on PCPartPicker. On ebay it looks like you can get a used 10400 for only a little more than a used 9600 (admittedly I don't typically buy used so I could be missing something there). I don't know about compatible motherboards though.
If you can't make it work for the B580 then a 2080 for $200 would offer excellent price to performance, though you may find yourself limited by the 8 GB VRAM before too long. If you can find a 6700 XT for no more than $260 or a 2080 Ti for no more than $275 that would offer about the same price to performance as the B580 (based on TechPowerUp's database of relative GPU performance). Knock about $30-40 off those prices to have price to performance on par with a $200 2080.
I have 9700k and a 1080. For my rig, the processor seems fine for now but is starting to show its age in some areas but a solid choice still. The 1080 is really showing its age and I'm considering getting a 4000 series soon. I would skip the 2000 series because from the 1000 it isn't much of a jump. Try looking at the 3000 line and good luck.
I think I'm gonna go for the 3060ti. It's better than the 4060, and can provide me some future proofing for my next build. This time around I'm only spending $90 on ram and a motherboard. I already have a 9600k sitting around.
So in a year or two I'll go with a more current gen ryzen 5 and I'll still have a 3060ti for it. I enjoy the budget, and cheap builds. Keeps me always wanting more at the end of the day. Keeps me greatful lol