Really the rx 560 was low end when it was first launched.
As far as the motherboard check the cpu support list. I’ve got an asrock ab350 pro 4 board that I’m using with a 5800x. Check the notes on the later bios versions for your board and see if they added that support. If so, something like a 5600x or 5700x would be a night and day difference from your 1600.
You are almost certainly going to need to upgrade the power supply. Probably a good 650 watt or better unit.
That said I’m of the opinion that the 3060 isn’t a very good value. For about the same money, in the USA at least, you can get the 6700xt which is closer to the 3070 and should blow the doors off a 3060. For less money you can look at the 6600, 6600xt, or 6650xt which are close to the same performance as a 3060 but cost less.
If you are willing to buy used and are a cheapskate, the other option is the 5700xt. I think these first came out in about 2019. Many of them have probably been mined on, but performance wise they are about the same as the rtx 3060/rx 6600xt without ray tracing at least. If you watch you can sometimes pick them up very cheap. Just for example, in the USA on eBay you can find ones like this for about $160 dollars plus shipping.
https://www.ebay.com/itm/1659782766...1QoO_yZSxW&var=&widget_ver=artemis&media=COPY
I actually picked up one around a month or so ago from a miner for under $150 shipped, but it works. For the money and getting 3060 performance it’s a good placeholder card and will tide me over until I can afford a better card later. So I may just use this card until other cards come out and see what he card I want. Who knows maybe I’ll keep it and make a little Xbox style pc for my TV and use steam big picture mode and use this in it for 1080p.