Yeah looks like a motherboard issue possibly, it happens all the time at my job. No HDD showing up in Windows 10 installation GUI is a big red flag, and if 3 short beeps is RAM failure, it definitely sounds like a bad motherboard.
Replacing this in a laptop I hear is really tricky, I've never done it before but I'm sure it can be done. Check to see if your laptop is still under warranty @Alcohol and if not look into how much replacing the motherboard yourself would be. From there, you need to determine your time/effort vs. money and decide if you want to replace the motherboard or buy a new unit.
Also one thing I saw in a photo above, your boot mode is set to UEFI. Try changing it to Legacy and see if that helps, it could have been changed for whatever reason or by accident. Yes, even with Windows 10.