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From the 580 to a 3070 ti or 6750 XT will give you about 3x performance increase. Don't worry about your screen ,you can always downsampling, I recommend get a 3070 ti if you care about ray tracing also I read somewhere that Nvidia card is better for VR, otherwise get a 6750 XT they both perform the same in rasterization.

What are your use cases? What is your current specs(CPU,RAM,PSU,GPU)?

That isn't how it work,silent bios will reduce fan speed, it quieter but not cooler.

Did you turn on ray tracing? If you want ray tracing and high fps, you need combine it with DLSS.

It seems like VRAM related problem,or overclock too high, ddu the driver on safe mode and hope it will back to normal. If it's still doing this try it on another PC before RMA your card.

Did you overclock your card? If so,change it back to stock profile and run furmark again. Edit: I see you ran furmark without overclock,what is your PSU? Maybe the PSU not provide enough power.

8 gb is not gonna enough for modern games,you need at least 16 gb. Anyway, upgrade from a 2060 to a 3060 won't give you that much performance,you should save a bit more then upgrade your ram to 16gb and get a 3060 ti instead.

Did you set your global setting power management to normal?

I don't understand how come you got a 5900x,a 3080 and still using 650 watt PSU? You need a new PSU immediately, otherwise it could potentially damage your component especially GPU and motherboard.

It's sad to hear that you had bad experience with them,I used to have one of their dual fan GTX 660 and it performs well,I even sold it to a friend after 3 year and it's still work now.

Is the card brand new? Maybe try to contact your local Gigabyte customer service and ask whether they can RMA it for you.

You want to upgrade to a 3060 ti from a 2070 super? How is your budget,if you can upgrade to a 3080 that's would be much better than the 3060 ti or even the 6750xt.

3060 ti is stronger than 2070 super,about the same as 2080 super.

Your GPU hot spot temp is high,it shouldn't be 20 degree celsius higher than the GPU. Like @PDifolco said, you should try to replace thermal paste first,if it's still high then maybe the fan of the GPU was stuck or running too low, increase its fan speed could solve the problem but it could be noisier.

Linus you should hire this guy immediately, let the man dream come true.