You have run across a problem which, to be fair, is not obviously mentioned in the data sheet, or at least not as you were thinking about it. That is the problem of input common mode voltage. First, keep in mind that the 741 was very much oriented toward the then-standard +/- 15 volt supplies which dominated at the time. Granted, the data sheet suggests that a 741 can operate down to +/- 5 volts, and your experiment with +/- 6 confirms that.
With that in mind, notice that the data sheet does not actually provide numbers for any power supply voltage other than +/- 15. Now look about halfway down and find "input voltage range", which will show a minimum of +/- 12. What this means is that, if your input voltage is less than 3 volts from either supply (for +/- 15), the IC is simply not guaranteed to work. Notice that this does not guarantee that the IC won't work, only that if it doesn't then it's not a concern of the manufacturer and you have no complaint. Is this 3 volt limit appropriate for a 12 volt or +/- 6 volt supply? Dunno. The data sheet doesn't say. You can play around with your +/- 6 volt setup and find out, but keep in mind that, just because an input closer than 3 volts works with the specific chip you have, there's no guarantee that it will work with another amp, particularly one from another manufacturer.
So, when your original setup provided a 0.5 volt input, that input was grossly out of bounds, and the result was as expected. When you went to +/- 6, the input was indeed within the proper range, and the circuit worked. You might want to play around and find out (just for your own amusement) just where the 741 stops working for different supply voltages. Just don't expect that another amp will behave exactly the same.