Don't you think that the on board components are actively accessing the EEPROM when it is powered? That would definitely interfere with another device trying to do the same thing.
Huge blunder to connect 5V to it without knowing the supply voltage. You may have destroyed the board already.
The fact that it still works does not mean that you have not damaged it. Excess voltages can weaken a device so it appears to work, but will fail later on. If you're not motivated enough to make a simple voltage measurement, I am leaving you on your own.
The chip is not responding to your attempt to detect it. Perhaps the chip is faulty. Perhaps the rest of the circuit is preventing you from detecting it in some way, like attempting to access it at the same time you are also attempting to access it. Either way, I think you have to remove the chip, it at least the Vcc pin.
I can see you soldered wires to the pins, so your solder skills are not as poor as you think. Do you have a solder sucker?
FYI it is unfortunately really easy to lift and destroy a pad while desoldering an SMT part. It does take some skill and extreme patience to do safely. The easy way, is to cut the leads off first but that destroys the part.