A friend's S3 had a damaged voltage regulator on the main board
If the hardware issue you're having with your phone has anything to do with the power-related parts on the main board, it is worthwhile to open the phone up and measure voltage rails with your multimeter. However, due to the extremely fine pitch of the galaxy s3's internals, it is a very difficult fix. The GS3's power comes from MAX77693 and MAX77686, which are both BGA devices. Capacitors surrounding the power chips should provide some accessibility to the power rails, though a quick google search did not reveal to me which capacitor is connected to what rail.
I strongly advice against trying to transplant the nand chip as it is (you guessed it) really fine-pitched BGA. However, if you have access to a working GS3, it is possible to carefully open it up, measure various parts to try and identify the power rails, and feed (for example) 3.3v directly to those components of your broken gs3. With a bit of luck, only the 3v3 rail is broken and you're able to temporarily revive your phone by putting it on life-support.
There are also companies that specialize in data-recovery from USB sticks which should have no issue getting the data off of your GS3, though they will charge extortionately high prices.