SpookyCitrus

That's due to a driver conflict from the old card's drivers. Anytime you switch GPUs you should remove the old graphics drivers before installing the new card. DDU(Display Driver Uninstaller) is a free program you can get from GURU3D.com that will automatically uninstall the driver for you. You also wouldn't of had to reset the PC either if you had done this, prior or after you started getting the crashing as it can be ran in safemode as well.
 
Just launch DDU, select GPU for the Device and AMD for the manufacturer and then click the remove and shutdown option. Once the computer shuts off after uninstalling you can unplug the power cable from your PSU and swap out the GPU, boot it back up and install the 4070's drivers/Geforce Experience and you'll be all set.

So you'd rather pay more, for less performance? The XBOX Series X runs at a minimum of 60fps and that's at 4K in fidelity mode. And performance mode in a lot of games is 120fps. I'd at least read up on it, for what games you're wanting to play and at what settings. I don't think you could get a computer that would run better for the price. https://www.xbox.com/en-GB/consoles/xbox-series-x 
 
If consoles just aren't your jam, I'd understand, but just having been in your shoes in the past, It's so much more worth it to save up and get a better computer from the get go than to go for the bare minimum. You'll just end up spending more in the long run. The console would just be a better alternative for your budget.

This is going to be the culprit, you're pairing an unlocked i7 with a budget B760 board. You're essentially crippling your CPU. I'd recommend getting a Z790 motherboard. If you are in the US the Z790 Gigabyte Eagle AX has been on sale for around $185 the last couple weeks. It's a solid board I've used in a couple of builds. Alternatively, there is the ASROCK Z790 Lightning which is pretty much the cheapest you can get a Z790 with WiFi built in if that's a must for you, I've had pretty solid experiences with ASROCK in recent years. If you'd like to see all of your options, use PCPartPicker and input the rest of your system components to ensure you find the right board.
 
https://pcpartpicker.com/product/Yj7scf/gigabyte-z790-eagle-ax-atx-lga1700-motherboard-z790-eagle-ax

Microsoft Flight Simulator and Forza Horizon 5 on Ultra? On a 6500XT and 12100f? Depending on your resolution you're not going to even get 30FPS. Fortnite in performance mode at medium/high settings or even ultra should be fine. I'd definitely look into either saving up some more money for a better system. Or going the used route, check FB marketplace or local classifieds sometimes people get rid of some wicked systems for cheap. Maybe getting the 12100f and pairing it with a used 30 series RTX card like a 3070?
 
What Calculator are you using for this? Most of these calculators are complete shams. Going off of YouTube videos of the specs you want playing these games which is actual performance compared to the generated stats of whatever website you are using, Cyberpunk is only getting an average of 40fps on low settings at 1080p. Ultra, there's no way that calculator is right.
 
With your budget and wants, maybe getting an XBOX Series X may be more within your grasp. 

This is going to be the culprit, you're pairing an unlocked i7 with a budget B760 board. You're essentially crippling your CPU. I'd recommend getting a Z790 motherboard. If you are in the US the Z790 Gigabyte Eagle AX has been on sale for around $185 the last couple weeks. It's a solid board I've used in a couple of builds. Alternatively, there is the ASROCK Z790 Lightning which is pretty much the cheapest you can get a Z790 with WiFi built in if that's a must for you, I've had pretty solid experiences with ASROCK in recent years. If you'd like to see all of your options, use PCPartPicker and input the rest of your system components to ensure you find the right board.
 
https://pcpartpicker.com/product/Yj7scf/gigabyte-z790-eagle-ax-atx-lga1700-motherboard-z790-eagle-ax

The only coolers that offer an LCD/Digital upgrade from standard are Corsair. Deepcool doesn't offer it as an option, and likely will not just sell you the digital cover. You'll need to just buy a separate cooler. 

With everything you have done, there isn't much left. I'd at least update the motherboard bios, you didn't mention trying that, I'd do that first.
If that doesn't work I'd say most likely next step is to test the system with a new PSU.

Looks good, only thing I would change is ditching the two mechanical HDDs for SSDs and doing an Nvme SSD instead of the Sata for the boot drive. If it's just for gaming there is literally no point in putting mechanical HDDs in the computer. Do a 1TB or 2TB Nvme SSD for your boot drive and then a 4TB Sata SSD for your storage drive.

Standard superglue and careful application should be all you need. Don't use a lot and only get it on the heat spreader and plastic RGB diffuser. You can get a gel style superglue as well so it's less runny and will stay in place.

Standard superglue and careful application should be all you need. Don't use a lot and only get it on the heat spreader and plastic RGB diffuser. You can get a gel style superglue as well so it's less runny and will stay in place.

It's frowned upon to double post so quickly
Sounds like a dying card. It *might* just need a vBIOS flash, but at this point, you really need to look at a replacement

What are the specs of the rest of the system? What are you trying to use it for? 

There is a small chance you lost the PSU lottery *twice*, but it's not that likely. I'd suspect something wonky going on with the mobo. 
 

though 5600 28 might be worth a gamble if you're the type of person that wouldn't mind taking time to tune the ram

Ahh, I see. Yeah, they're not very common and usually a lot more expensive.
Nvm guess they aren't that much more anymore. The last time I looked into sub 30CL DDR5 it was $40 more or higher.

You can use internet recovery, a program such as Mint to make an install USB for MacOS(you can also do this with an installer download), similar to making one for Linux or Windows. As long as you have a compatible version of the OS that is supported by the Mac in question. You'll need to either install the MacOS version that originally came on the Mac or one that is still supported by it. Apple cuts support for certain Macs so that they won't work with newer versions of MacOS. You can figure out the latest supported version your Mac will work with by looking up the serial number on everymac.com

Nvme SSDs are designed to work without a heat spreader, so either one is going to work. Typically you will need to remove the preinstalled one in order to use the motherboards built in. Usually the built in MB ones are better. Personally, I order Nvmes without the heat spreader or if the heat spreader version is cheaper I remove it and use the motherboard's if it has one. 
 
Usually you order the with or without versions based on if your motherboard has one or not. In this instance where your board has one already, you'd want to get an Nvme that doesn't already have a heat spreader. Regardless, even if it didn't come with one or the motherboard didn't have one you'd still be fine. 

Nvme SSDs are designed to work without a heat spreader, so either one is going to work. Typically you will need to remove the preinstalled one in order to use the motherboards built in. Usually the built in MB ones are better. Personally, I order Nvmes without the heat spreader or if the heat spreader version is cheaper I remove it and use the motherboard's if it has one. 
 
Usually you order the with or without versions based on if your motherboard has one or not. In this instance where your board has one already, you'd want to get an Nvme that doesn't already have a heat spreader. Regardless, even if it didn't come with one or the motherboard didn't have one you'd still be fine. 

This is most likely going to be a problem with the new board/CPU, possibly the APU causing some sort of issue with the dGPU being able to output through the board when it's in use, or something else entirely. Those mining cards are not meant to be used like this and require tweaking software or specific hardware to work like that. Most likely you will only be able to use it in that way with an Intel based system. 
 
You can try disabling the iGPU in device manager/bios and that should in theory let it only use the dGPU for output but those mining cards are iffy. If you disable it you might not be able to get display at all anymore which would let you know that it's not going to work the way you want in that configuration.
 
If I were in this predicament, personally, there are so many inexpensive standard dedicated GPUs out there that are much better than that mining card, I wouldn't even mess with it. I'd cut my loses and buy a used RX 580(super inexpensive) or go for a used RTX 20 or 30 series. Hell, the new RX cards aren't super expensive either. 

Not trying to derail your build thread, BUT...
 
If you want to upgrade to Windows 11 (which sounds like the sole goal here) without spending any money, it's fairly easy to do. You can modify the installation ISO with a script or RUFUS to bypass those hardware checks. (I usually use the official ISO then modify it with this tool). I have done it on dozens of "unsupported" systems with no ill effect.

Use WizTree or Windirstat to view all files, remove temporary or unnecessary files it finds that take up a decent amount of space. 
 
If it were my computer I'd just buy a new drive, 1TB sata SSDs, and Gen 3 Nvmes are pretty inexpensive these days. Like @PDifolco said, it's pretty pointless to keep constantly cleaning a 250GB.

A format and reinstall is a last resort, this could still just be a simple driver/software fix that doesn't require a format and reinstall. We've already ruled out a cryptominer virus. I'd advise against going this route until we do some more troubleshooting. 

The cheapest board I'd be willing to tolerate with that chip would be the Z790-A Pro.
https://uk.pcpartpicker.com/product/xNKKHx/msi-pro-z790-a-wifi-ddr4-atx-lga1700-motherboard-pro-z790-a-wifi-ddr4

Budget?
Something like the Yeti Nano or Seiren Mini come with a stand to sit in front of you. 
 
Those microphones are meant for conferencing, and are notoriously poor for sound quality. 

Get a bigger drive anyway, you will waste your life trying to clean a 250GB drive ...
 
A Gen3 1TB NVMe is $60, less than a console game

None of the 7800X3D systems I have worked on or put together use the same board that he has, his system was purchased and researched by himself, all I did was assemble it. I tend to find a good board and stick with it, all of the 7800X3D systems I have done use the same board and its a different model from what he used. His is also a very difficult board to work with, it arrived with a dead CMOS battery(a lot of this model shipped with dead or dying batteries, Gigabyte already released a statement) and essentially the entire board needs to be taken apart to change it, but that's a separate issue that was already resolved. If you're curious he has a Gigabyte X670E Aorus Master.