I've tried 575, 575-open, 570 and 570-open drivers

Who doesn't recoginize the card . . . . Boinc or the GPUGrid apps?

Sorry, I have no experience with Linux Mint.

This worked for me at equivalent Ubuntu Linux kernel, NVIDIA RTX 40XX series.
From Terminal window:

sudo apt install software-properties-common -y
sudo add-apt-repository ppa:graphics-drivers/ppa -y
sudo apt update
sudo apt install nvidia-driver-570

I don't know if this could be expected to work at Linux Mint also...

Keith - Nothing sees it yet.

Thanks Service! I will try that.

Well I assume something is driving a display for you to run the OS. I guess you are running off the HD4000 igpu in your cpu then?
What does lspci | grep VGA output show for detected graphics devices on the host?
What motherboard are you using?
Somewhere in the BIOS there is a place to select using the internal gpu or a PCIE device. What have you selected there?

I ran out of time in Hawaii. Short weed pulling trip. Yes, i have to prefer the igpu to let nvda run cuda optimally. The old ASRock Z77 extreme4 works fine with the 4060ti's and will probably to be safe bring them back out of the win box in late October. I'm doing a new build with a gigabyte b650 eagle ax. If the 5060ti's work in that, I'll get 2 more for the win box.

Those 4 commands worked on the new box. Not at first as nvidia-smi showed no devices but after rebooting and changing the bios to say prefer IGPU (gigabyte used a different acronym) nvidia-smi showed the GPU. Added a 2nd one and it's showing too.

I see a ton of "ubuntu" statements while installing drivers and updates. It seems that mint is a very close relative. I was this close (fingers pinched) to switching to straight ubuntu.

It's working on amicable at the moment. We'll see when the new box gets a grid task.

Thanks again Mr Science.

BTW, at first blush, the 5060Ti MSI ventus is about 14% faster than the 4060Ti (also msi ventus). Not a perfect comparison because different MOBO and OS.

Mint is just a Ubuntu clone with a different graphical front end.

That helps a lot. It was a mystery to me before. Mint is the same cake with different frosting.

I had another hiccup. Kept getting these weird messages at boot. Pci 0000:0f:00.0: Failed to read data

Next line: hub 8-0:1.0: config failed; hub doesn't have any ports! (err -19)

Driving me nuts as it wouldn't boot hanging there.

Kept searching then some post suggested checking kernel. That was it. Even though I installed 22.1 I had a pretty old kernel. Even update manager said I was all up-to-date when I wasn't.

Mint usually is about a full release behind Ubuntu usually. Glad you sorted it out. The error message about the missing USB hub is common but shouldn't have prevented booting. It is just observational. But the PCI error was likely the reason for not booting.
The latest beta Mint (Zara) lines up with the latest Ubuntu which is using the 6.14 kernel now. That happens to be a LTS release also so once it comes out of beta you probably should update to that.

I had similar problem. My PC is dual boot Windows/Linux. The solution was with MBR/GPT. If you have dual boot too, your drive needs to be formated in GPT.

For this setup it was the old kernel that kept me from booting. To get the 2nd 5060 to work I had to go into the bios and change pcie speeds from auto to the max which is gen4. I also changed the pcie bifurcation to 8x8 but I don't know if that was material.

Spinning like a top for the last 36 hours.

Room on mobo for 4 GPUs. Each 5060 Ti is running at 100% at 106 and 114W.

Kudos for sticking with it and attacking the issue methodically. Keep crunchin'

Mint (not LMDE) is always tied to the Ubuntu's most recent LTS release.