The HDD is a WD80EAAZ - 00BXBB0 manufactured in Feb 2024. It's the odd 5640 rpm version.
PSU is a Seasonic Focus Gold 750W. Was purchased and in use since September 2018. I haven't noticed any power issues. My overclocked CPU and undervolted GPU have remained perfectly stable.
Motherboard is a MSI Pro X670-P WIFI.

I bought a new WD Blue 8TB HDD in May (shipped and sold by Amazon). I was going to use it as an external, but I installed it in my computer temporarily so I could copy files to it easier. I filled about 4TB and then procrastinated about removing it and putting it in an external enclosure. It was still in my computer when the power went out to my house for about 1 second and turned back on. I unplugged my computer and waited a couple hours to make sure the power was stable before turning my computer back on again. When I plugged my computer back in, it wouldn't boot and I could hear the HDD spinning up and making a beeping noise over and over. I knew what this meant, so I shut it down and removed the drive.

The computer booted fine and I plugged the HDD in with a USB adapter so I could record the noise it is making. You can hear the platters spin up and then I think there might be the sound of the heads moving before hearing something that sounds like two beeps, then the process repeats. I uploaded the audio here:

You will need to turn your volume way up to hear it spin up. It goes through two cycles. The "beeps" are the loudest sounds at around 6s and 15s.

https://www.youtube.com/watch?v=pfoqWzmZ00o

Unfortunately I do not have another copy of most of the files. I would like to be able to recover the data if possible, but it's not important enough to me to pay hundreds of dollars for professional data recovery. I have learned my lesson and if I am able to recover the data, I will get two drives to have two copies of everything. I am also thinking about getting a UPS for my computer.

I was told to check the TVS diodes. I checked the two larger chips on the bottom left labeled 6BG 28A and 6AE 25A. I checked multiple times and both gave the exact same readings. Putting the black probe on the side with the line gave a reading of 149 on the diode test setting or 83.5 when using the 200 ohm setting. Reversing the probes so the red one is on the side with the line does not give a reading on the diode test or 200 ohm settings. I believe that means that they are working, but I don't know if they are working properly because I don't know what the readings should be.

Does anyone know what would have failed to cause this specific beeping type sound? Maybe there was some kind of power surge that burned something out on the board? I did find out recently that I can buy a replacement board for $50 and have the company swap the BIOS chip for me, but I would like to diagnose what failed if possible so I don't end up wasting the $50 just to find out that it is a more serious issue that will cost more than I am willing to spend to recover the data. I don't have a lot of knowledge about electrical components, but I have a multimeter and can follow instructions if anyone has any suggestions.

The pictures are too large to attach, so I will link them.
https://i.imgur.com/OcuxL1C.jpeg
https://i.imgur.com/x8THmmO.jpeg

Maybe heads stuck to the platter?

Not heads stuck, that's for sure.
That's the sound of heads clicking. Probably the preamp got toasted with the power outage.
Need to take it to a proper lab.

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From a quick search, it sounds like the preamp is inside the drive on the head stack. That will cost more to replace than I am willing to spend.

Is there a way I can test to verify if it's the preamp or something else that failed? If I can verify it as being something on the board, I am willing to pay the have the board replaced since it's fairy cheap. If it's an internal issue, I will just RMA the drive and not recover the data.

Would it be helpful to try and get a louder, more clear audio recording? I haven't plugged the drive back in yet, but if I am going to RMA it and lose the data anyway, risking the data by letting it spin up doesn't really matter.

The power outage could have resulted in the drive crupting its system files hidden in the disk, G-List and so so, so the drive cannot load. I don't understand why you don't have the original files on your original disk if you were just copying. Perhaps if you moved the files, you can recover them from the original disk, then send the failed drive for warranty and then get a new one.

Some small files are backups and I do have the originals on my boot SSD, but I don't have copies of most of the other files. I had a bunch of old small HDDs and wanted to consolidate them. I ran a full surface scan on the new 8TB when I got it and then copied the files from the small drives to the new 8 TB. Since everything was working fine, after about 3 months, I formatted the old small drives.

I don't know why I didn't think of this before, but I haven't written anything to those drives I formatted that I was copying the files from, I just did a quick format on them. I have never tried to recover data from a formatted drive, but I'm guessing there is free software I can use. I haven't searched yet, but maybe someone here has a recommendation?

If you just formatted and did write on them chances are good.
Take a look at DMDE (cheap) or even Recovery Explorer: https://www.r-explorer.com/license/

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Try recovering the files if you can, a quick format will not destroy the main data on the disk, but when you recover, make sure you clone the original disk, and don't write anything to it. Recover files to another disk.

You may get most of your stuff back, but you may also loose some. Good luck.